#6554752

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533087
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1364 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fi,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,c^b=ch,g^c=gi,e^d=eh

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6461948);