#72233181

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#23984
commutator, Frattini, cosocle	C₂³×D₄	C₂⁵
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂²×D₄² ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

At least 9520 conjugacy classes of subgroups; only the 666 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^11|b^2=h,c^2=h,b^a=bh,c^a=ch,c^b=chj,d^a=df,d^b=dfk,d^c=dfg,e^a=ek,e^b=ei,e^c=ek,f^b=fi,f^c=fij,g^a=gjk,g^b=gij,g^c=gjk,g^d=gk,g^f=gk,h^d=hi,j^d=jk

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233213
maximal normal subgroup of	#72233213

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