#501808

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

Polycyclic presentation:

2^9|b^2=fi,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, 409004);