#6676257

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1958 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,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, 6583453);