#6699485

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
prime radical	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1473 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=fg,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, 6606681);