#72232684

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#23900
commutator, Frattini, cosocle	C₂²×C₂²⋊C₄	C₂⁵
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×#120 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

At least 6255 conjugacy classes of subgroups; only the 772 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:

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

Polycyclic presentation:

2^11|a^2=ik,d^2=ik,f^2=jk,d^a=dg,d^b=dg,d^c=dik,e^a=eh,e^d=ef,f^a=fi,f^b=fj,f^c=fk,f^d=fj,f^e=fjk,g^e=gj,h^a=hk,h^d=hijk,h^f=hk,i^e=ik

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234842, #72237971
maximal normal subgroup of	#72234842, #72237971

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