#72321542

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₈
Frattini, cosocle	C₈²	C₂⁵
socle	C₂²	C₂²×#1580
prime radical	#7623986	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#131 ◁ #479903 ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

3247 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 64 points:

(33,34)(35,47)(36,52)(37,56)(38,44)(39,59)(40,42)(41,49)(43,51)(45,54)(46,53)(48,63)(50,61)(55,64)(57,60)(58,62);(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)(33,35,37,48,39,50,38,57,40,51,54,36,53,55,58,49)(34,41,44,61,46,52,56,47,42,60,62,64,59,63,45,43);(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)(34,42)(41,60)(43,47)(44,62)(45,56)(46,59)(52,63)(61,64);(3,11)(7,17)(12,23)(13,24)(18,27)(19,28)(25,31)(29,32);(33,36,54,55,39,49,58,35,40,48,37,50,53,57,38,51)(34,43,44,60,46,61,56,63,42,47,62,41,59,64,45,52)

Polycyclic presentation:

2^11|b^2=f,e^2=gi,f^2=h,g^2=ik,h^2=j,i^2=k,b^a=bg,c^a=ck,d^c=djk,e^a=eg,e^b=ei,f^a=fi,f^e=fk,g^a=gi,g^b=gk,h^a=hk,i^a=ik

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322318, #72323169
maximal normal subgroup of	#72322318, #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, 2, F.6);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.11);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.10*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);