#6657925

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632082
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1397 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=f,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=dfhi,g^c=gi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6565121);