#6818241

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806551
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1693 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313243, #72313674, #72313880
maximal normal subgroup of	#72313243, #72313674, #72313880
maximal quotient of	#72320003

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6725437);