#6806003

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682420
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

1222 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:

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

Polycyclic presentation:

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

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, 6713199);