#6690915

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667508
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1311 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=g,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,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, 6598111);