#72322475

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#1333
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	#1311	C₂⁴
prime radical	#7623986	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#131 ◁ #479943 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1589 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 96
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 96 points:

(1,2,4,8,5,9,14,20,6,10,15,21,16,22,26,30)(3,7,12,18,13,19,25,29,11,17,23,27,24,28,31,32)(33,34,37,42,36,44,53,45,39,46,58,73,55,41,68,69)(35,47,49,79,51,78,85,80,52,81,86,74,48,82,56,93)(38,59,61,43,54,75,87,76,62,94,57,77,92,72,91,70)(40,60,64,90,63,84,71,83,66,95,96,50,67,88,65,89);(33,35)(34,40)(36,48)(37,56)(38,60)(39,52)(41,63)(42,71)(43,74)(44,67)(45,64)(46,66)(47,75)(49,68)(50,87)(51,55)(53,86)(54,88)(57,83)(58,85)(59,78)(61,89)(62,95)(65,73)(69,96)(70,93)(72,81)(76,80)(77,79)(82,94)(84,92)(90,91);(1,3)(2,7)(4,12)(5,13)(6,11)(8,18)(9,19)(10,17)(14,25)(15,23)(16,24)(20,29)(21,27)(22,28)(26,31)(30,32)(33,36,39,55)(34,41,46,44)(35,48,52,51)(37,53,58,68)(38,54,62,92)(40,63,66,67)(42,69,73,45)(43,70,77,76)(47,78,81,82)(49,56,86,85)(50,83,90,89)(57,91,61,87)(59,72,94,75)(60,88,95,84)(64,71,96,65)(74,93,79,80);(3,11)(7,17)(12,23)(13,24)(18,27)(19,28)(25,31)(29,32)

Polycyclic presentation:

2^11|a^2=e,c^2=k,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,b^a=bf,c^a=ck,d^c=dik,e^b=ejk,f^a=fh,f^b=fhjk,f^e=fk,g^b=gk,h^a=hj,h^b=hjk,j^a=jk,j^b=jk

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322541, #72323169
maximal normal subgroup of	#72322541, #72323169

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.11);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.10*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);