#6556901

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1370 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)(11,19)(12,21)(13,17)(14,16)(15,18)(20,23)(22,24)(25,26,31,34)(27,29,33,30)(28,35,37,36)(32,39,40,38);(2,5)(6,8)(9,11,15,14)(10,16,18,19)(12,22,13,20)(17,24,21,23)(25,27,30,26,31,33,29,34)(28,36,39,40,37,35,38,32);(1,3,4,7)(2,6,5,8)(9,12,15,13)(10,17,18,21)(11,20,14,22)(16,23,19,24)(25,28,30,39,31,37,29,38)(26,32,33,36,34,40,27,35);(9,13,15,12)(10,17,18,21)(11,20,14,22)(16,24,19,23)(28,37)(32,40)(35,36)(38,39);(9,14,15,11)(10,16,18,19)(12,22,13,20)(17,23,21,24)(25,29,31,30)(26,33,34,27)(28,38,37,39)(32,36,40,35)

Polycyclic presentation:

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

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, 6464097);