#6763321

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1341 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)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,30,32)(27,36,37,40)(28,39,33,34)(29,35,31,38);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(26,32)(27,37)(34,39)(35,38);(2,6)(3,7)(25,27,29,34,30,37,31,39)(26,33,38,40,32,28,35,36);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,28,30,33)(26,34,32,39)(27,38,37,35)(29,36,31,40);(3,7)(5,8)(11,19)(14,17)(18,24)(22,23)(25,29,30,31)(26,35,32,38)(27,34,37,39)(28,40,33,36)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,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, 6670517);