#6825548

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6732744);