#6854681

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1204 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=fghi,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,e^d=ei

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, 6761877);