#6693551

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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 ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1345 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)(9,10)(11,13)(12,15)(14,16)(17,18,24,23)(19,25,30,32)(20,22,26,21)(27,29,28,31);(1,3,5,4)(2,6,7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,25)(20,31)(21,28)(22,27)(23,32)(24,30)(26,29);(1,4,5,3)(2,6,7,8)(10,11)(14,15)(17,20)(18,22)(19,27)(21,23)(24,26)(25,29)(28,30)(31,32);(9,12,13,16)(10,14,11,15)(17,21,24,22)(18,26,23,20)(19,28,30,27)(25,29,32,31);(17,22,24,21)(18,26,23,20)(19,29,30,31)(25,28,32,27)

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,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, 6600747);