#6686935

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2027 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^d=ei

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, 6594131);