#6817588

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
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1301 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,15,16,13)(17,18)(19,29)(20,30)(21,28)(22,26)(23,27)(24,31)(25,32);(1,3)(2,5)(4,8)(6,7)(17,19)(18,24)(20,25)(21,30)(22,32)(23,31)(26,28)(27,29);(9,11,12,16)(10,13,14,15)(17,20,32,24,23,26,21,29)(18,22,25,19,27,30,28,31);(2,6)(5,7)(17,21,23,32)(18,25,27,28)(19,30,31,22)(20,29,26,24);(3,8)(5,7)(17,22,21,31,23,30,32,19)(18,26,28,24,27,20,25,29)

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=i,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6724784);