#6640530

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636940
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	#37570	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1314 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)(9,10,14,17)(11,18,20,15)(12,16,13,22)(19,24,21,23)(26,28)(27,34)(29,31)(32,39)(35,38)(36,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,18)(22,24)(25,26)(27,32)(28,30)(29,37)(31,33)(34,40)(35,36)(38,39);(3,7)(5,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,27,31,34,30,35,29,38)(26,32,33,40,28,36,37,39);(3,7)(5,8)(10,17)(11,20)(16,22)(19,21)(25,28,30,26)(27,32,35,36)(29,33,31,37)(34,40,38,39);(2,6)(5,8)(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(25,29,30,31)(26,33,28,37)(27,34,35,38)(32,39,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6547726);