#6700287

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1796 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6607483);