#6695886

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

1829 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16);(1,3,4,8)(2,5,6,7)(17,18,22,26)(19,27,29,31)(20,24,21,25)(23,30,32,28);(10,13)(14,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,31,32,27);(3,8)(5,7)(18,25)(19,29)(20,21)(23,31)(24,26)(27,32)

Polycyclic presentation:

2^9|b^2=fi,d^2=gi,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, 6603082);