#6647963

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632008
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1576 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)(11,18)(12,16)(13,21)(14,17)(15,20)(19,24)(22,23)(25,26,29,33)(27,34,30,31)(28,36,38,39)(32,37,40,35);(2,5)(3,7)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(25,27,29,30)(26,31,33,34)(28,37,38,35)(32,39,40,36);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,28,29,38)(26,32,33,40)(27,35,30,37)(31,36,34,39);(2,5)(6,8)(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(26,33)(31,34)(32,40)(36,39);(10,17)(15,18)(16,21)(23,24)(25,27,29,30)(26,34,33,31)(28,35,38,37)(32,39,40,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=f,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6555159);