#72266945

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#394419
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	C₂×#3532	C₂²
LCS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

3882 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^10|a^2=e,b^2=hij,d^2=hij,b^a=beh,d^a=dg,d^b=dhij,d^c=df,e^b=ej,f^a=fi,g^c=gij,g^d=gj,g^f=gj,h^a=hj,h^d=hj,i^d=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266950, #72266951, #72323233
maximal normal subgroup of	#72266950, #72266951, #72323233

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