#6696899

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667484
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 ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1678 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)(10,14)(11,12)(17,18,23,22)(19,24,28,31)(20,21,25,30)(26,27,32,29);(9,10)(11,15)(12,16)(13,14)(17,19,23,28)(18,24,22,31)(20,29,25,27)(21,26,30,32);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,23,25)(18,21,22,30)(19,26,28,32)(24,27,31,29);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,18,23,22)(19,24,28,31)(20,21,25,30)(26,27,32,29);(10,14)(15,16)(17,21,23,30)(18,25,22,20)(19,27,28,29)(24,32,31,26)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,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,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6604095);