#6690412

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	#35728	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1832 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,h^b=hi,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, 6597608);