#500461

order	512
nilpotency class	4
isoclinism	isoclinic class of #499736
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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

777 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)(3,7)(4,5)(6,8)(9,10)(11,20)(12,17)(13,16)(14,19)(15,22)(18,26)(21,37)(23,30)(24,36)(25,31)(27,33)(28,32)(29,39)(34,40)(35,38);(2,5)(6,7)(9,11)(10,15)(12,22)(13,23)(14,21)(16,31)(17,20)(18,30)(19,35)(24,28)(25,26)(27,38)(29,32)(33,37)(34,36)(39,40);(10,16)(11,21)(13,26)(14,28)(15,29)(17,18)(19,33)(20,36)(22,38)(23,39)(24,25)(27,40)(30,37)(31,35);(1,3,4,8)(2,6,5,7)(10,17)(15,20)(16,18)(19,33)(29,36)(30,31)(32,34)(35,37)

Polycyclic presentation:

2^9|d^2=ei,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407657);