#501985

order	512
nilpotency class	4
isoclinism	isoclinic class of #500730
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28247	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

533 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)(3,5)(4,6)(7,8)(9,10)(11,21)(12,26)(13,29)(14,18)(15,19)(16,20)(17,24)(22,36)(23,37)(25,33)(27,35)(28,38)(30,32)(31,34)(39,40);(1,3,4,7)(2,5,6,8)(9,11,15,31,16,25,13,24)(10,17,29,33,20,34,19,21)(12,23,27,40,28,36,14,30)(18,22,38,39,35,37,26,32);(2,6)(5,8)(9,10)(11,22)(12,18)(13,19)(14,26)(15,29)(16,20)(17,30)(21,36)(23,33)(24,32)(25,37)(27,38)(28,35)(31,39)(34,40);(9,12,15,27,16,28,13,14)(10,18,29,38,20,35,19,26)(11,23,31,40,25,36,24,30)(17,22,33,39,34,37,21,32)

Polycyclic presentation:

2^9|b^2=efi,d^2=ei,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gh,h^b=hi,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, 409181);