#6584659

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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
prime radical	#25115	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1923 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,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, 6491855);