#499069

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3155	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1292 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)(3,4)(6,7)(11,20)(12,14)(13,25)(16,21)(17,29)(18,26)(22,39)(23,36)(24,28)(27,30)(31,34)(32,37)(33,38)(35,40);(5,6)(7,8)(9,10)(11,21)(12,17)(13,26)(14,29)(15,19)(16,20)(18,25)(22,37)(23,38)(24,35)(27,31)(28,40)(30,34)(32,39)(33,36);(2,4)(5,7,8,6)(9,11)(10,16)(12,22)(13,27)(14,30)(15,23)(17,31)(18,34)(19,33)(20,25)(21,26)(24,36)(28,39)(29,32)(35,37)(38,40);(9,12,15,14)(10,17,19,29)(11,22,23,30)(13,24,28,25)(16,31,33,32)(18,26,35,40)(20,27,36,39)(21,37,38,34)

Polycyclic presentation:

2^9|c^2=e,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406265);