#6704379

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1235 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6611575);