#72231961

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂³×C₄²
commutator	C₂×C₄	C₂³×C₄×C₈
Frattini, cosocle	C₄×C₁₆	C₂⁵
socle	C₂²	C₂³×#346
prime radical	#3444	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 80 points:

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

Polycyclic presentation:

2^11|d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,b^a=bi,c^a=ck,d^b=dk,e^d=ej,f^e=fk,g^d=gk

Conjugacy classes & character table

416 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231972, #72231993
maximal normal subgroup of	#72231972, #72231993

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, 4, F.6);;
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.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);