#72225267

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#72216174
commutator	C₄×C₁₆	C₂³×C₈
Frattini, cosocle	C₄×#346	C₂⁴
prime radical	#72216275	C₄
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×C₈² ◁ #72216161 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3045 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(2,5)(6,8)(10,15,18,20,19,30,21,31)(11,22,23,25)(12,26)(16,28,32,29,33,39,37,40)(17,34,35,36)(24,38);(2,6)(3,7)(5,8)(10,16)(11,17)(12,24)(13,27)(15,28)(18,32)(19,33)(20,29)(21,37)(22,34)(23,35)(25,36)(26,38)(30,39)(31,40);(9,10,17,28,12,20,22,37,27,18,36,40,38,30,23,33,14,19,35,39,26,31,25,32,13,21,34,29,24,15,11,16);(1,2,7,8,4,5,3,6)

Polycyclic presentation:

2^12|a^2=e,c^2=fi,d^2=gj,e^2=h,f^2=ik,g^2=j,i^2=kl,k^2=l,c^a=cik,c^b=cf,d^a=dj,d^b=dg,e^c=ek,f^a=fkl,f^b=fi,f^e=fl,g^b=gj,h^c=hl,i^a=il,i^b=ik,k^b=kl

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225288, #72227231, #72227291
maximal normal subgroup of	#72225288, #72227231, #72227291
maximal quotient of	#72225268, #72225288

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.6*F.9);;
gap> SetPower(coll, 4, F.7*F.10);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9*F.11);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 9, F.11*F.12);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.11*F.12);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.12);;
gap> SetConjugate(coll, 9, 1, F.9*F.12);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 11, 2, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);