#6722272

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667836
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	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1466 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)(3,8)(4,6)(5,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,29,31)(27,35,37,32)(28,38,36,33)(30,34,39,40);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(26,31)(27,36)(28,37)(30,39)(32,33)(35,38);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,27,29,37)(26,32,31,35)(28,39,36,30)(33,40,38,34);(10,16)(14,17)(15,22)(21,23)(26,31)(27,37)(28,36)(34,40);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(21,23)(25,28,29,36)(26,33,31,38)(27,30,37,39)(32,34,35,40)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fgi,e^2=g,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,e^c=ei,h^c=hi,e^d=ei,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, 6629468);