#512066

order	512
nilpotency class	4
isoclinism	isoclinic class of #511755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#32942	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,h^a=hi,f^c=fh,h^c=hi,g^d=gi

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, 419262);