#6693569

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667836
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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1395 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)(17,18,21,22)(19,23,26,25)(20,24,29,30)(27,31,32,28);(1,3,5,4)(2,6,7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,23)(20,28)(21,26)(22,25)(24,27)(29,31)(30,32);(1,4,5,3)(2,6,7,8)(10,11)(14,15)(18,22)(19,25)(23,26)(24,30)(27,28)(31,32);(9,12,13,16)(10,14,11,15)(19,26)(23,25)(27,32)(28,31);(17,20,21,29)(18,24,22,30)(19,27,26,32)(23,31,25,28)

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,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, 6600765);