#6693318

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1408 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=fg,d^2=f,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, 6600514);