#6584090

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2229 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491286);