#500711

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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×#3280	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1154 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:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=i,d^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=di,f^c=fh,g^c=gh,h^c=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, 407907);