#6679158

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667460
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1553 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,28,31)(27,33,36,30)(29,32,37,40)(34,39,35,38);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,28,36)(26,30,31,33)(29,34,37,35)(32,38,40,39);(2,6)(5,8)(27,34)(29,37)(30,38)(32,40)(33,39)(35,36);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(26,31)(30,33)(32,40)(38,39);(11,19)(14,17)(18,24)(22,23)(27,35)(29,37)(30,39)(32,40)(33,38)(34,36)

Polycyclic presentation:

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

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, 6586354);