#6614673

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1431 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,7)(3,8,4,6)(9,10)(11,14)(12,15)(13,16)(17,18,22,26)(19,27,29,43)(20,24,32,40)(21,34,35,25)(23,36,38,30)(28,42,31,46)(33,48,44,41)(37,47,39,45);(9,11)(10,13)(12,16)(14,15)(17,19,21,30,22,29,35,36)(18,23,25,27,26,38,34,43)(20,31,33,47,32,28,44,45)(24,39,41,46,40,37,48,42);(1,3,5,4)(2,6,7,8)(11,16)(13,14)(17,20,22,32)(18,24,26,40)(19,28,29,31)(21,33,35,44)(23,37,38,39)(25,41,34,48)(27,42,43,46)(30,45,36,47);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,16)(13,15)(20,32)(24,40)(28,31)(33,44)(37,39)(41,48)(42,46)(45,47);(11,16)(13,14)(19,29)(23,38)(27,43)(28,31)(30,36)(37,39)(42,46)(45,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=fhi,c^2=gi,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,e^d=ehi

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, 6521869);