#6693898

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	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1319 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=g,d^2=i,e^2=f,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, 6601094);