#72234097

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#393934
commutator, Frattini, cosocle	C₂²×C₄×D₄	C₂⁴
LCS	G ▷ C₂²×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2290 ◁ G
LECS	G ▷ C₂²×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

At least 7920 conjugacy classes of subgroups; only the 320 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^11|d^2=k,g^2=j,b^a=bh,c^a=ck,c^b=chik,d^a=dg,d^b=df,d^c=de,e^a=ei,e^b=eij,f^c=fi,f^e=fk,g^a=gj,g^b=gj,g^c=gik,g^d=gj,g^e=gk,h^d=hj,i^d=ik

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234105
maximal normal subgroup of	#72234105

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