#6569024

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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

1293 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,c^2=fi,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^c=ei,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6476220);