#501804

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
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	#24290	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

654 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)(11,18)(12,14)(13,17)(15,20)(16,21)(19,26)(22,35)(23,30)(24,33)(25,36)(27,31)(28,29)(32,34)(37,38)(39,40);(1,3,4,7)(2,5,6,8)(10,17)(11,22)(13,15)(14,19)(18,23)(20,21)(24,40)(25,37)(26,28)(29,31)(30,32)(33,35)(34,38)(36,39);(2,6)(5,8)(9,11,16,25)(10,18,21,36)(12,23,27,39)(13,24,15,32)(14,30,31,40)(17,33,20,34)(19,22,29,37)(26,35,28,38);(9,12)(10,19)(11,23)(13,28)(14,17)(15,26)(16,27)(18,22)(20,31)(21,29)(24,38)(25,39)(30,33)(32,35)(34,40)(36,37)

Polycyclic presentation:

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

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, 409000);