#6765130

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682340
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1451 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)(3,5)(4,6)(7,8)(9,10)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24)(25,26,30,33)(27,31,29,38)(28,35,37,40)(32,34,39,36);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,18)(21,24)(25,27,30,29)(26,31,33,38)(28,34,37,36)(32,35,39,40);(2,6)(3,7)(25,28,29,36,30,37,27,34)(26,32,31,35,33,39,38,40);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24);(3,7)(5,8)(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6672326);