#6672935

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667748
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60756	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1980 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,h^e=hi

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, 6580131);