#6567583

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1949 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=f,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6474779);