#6695367

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1772 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:

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

Polycyclic presentation:

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

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, 6602563);