#72240297

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1790
commutator	C₂×C₂²⋊C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂⁴
prime radical	C₂×#46710	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3924 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|a^2=h,b^2=ij,d^2=efgij,f^2=i,g^2=j,b^a=bij,d^a=de,d^b=df,d^c=di,e^a=ej,e^b=eh,e^d=ej,f^a=fhj,f^b=fi,f^e=fj,g^a=ghj,g^b=gh,g^d=gj,g^e=gj,g^f=gj,h^d=hj

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240308
maximal normal subgroup of	#72240308
maximal quotient of	#72240581

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