#418529

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1672 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=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, 325725);