#6696825

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667460
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

1735 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)(17,18,22,26)(19,23,20,24)(21,25,29,32)(27,30,28,31);(9,10)(11,15)(12,16)(13,14)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,19,22,20)(18,23,26,24)(21,28,29,27)(25,31,32,30);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15);(10,14)(15,16)(17,20,22,19)(18,24,26,23)(21,27,29,28)(25,30,32,31)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6604021);