#6556920

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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

1463 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)(26,30)(27,33)(28,35)(31,34)(32,38)(36,39);(2,5)(6,8)(9,11,15,14)(10,16,18,19)(12,22,13,20)(17,24,21,23)(25,26,28,30,29,32,35,38)(27,34,33,40,36,31,39,37);(1,3,4,7)(2,6,5,8)(9,12,15,13)(10,17,18,21)(11,20,14,22)(16,23,19,24)(25,27,35,39,29,36,28,33)(26,31,38,40,32,34,30,37);(9,13,15,12)(10,17,18,21)(11,20,14,22)(16,24,19,23)(26,32)(30,38)(31,34)(37,40);(9,14,15,11)(10,16,18,19)(12,22,13,20)(17,23,21,24)

Polycyclic presentation:

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

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, 6464116);