#505182

order	512
nilpotency class	4
isoclinism	isoclinic class of #500730
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#576	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

653 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,24)(13,23)(14,20)(16,19)(17,21)(18,28)(22,29)(25,34)(26,35)(27,37)(30,32)(31,38)(33,40)(36,39);(2,5)(6,8)(9,11,15,10,17,20,14,21)(12,25,28,36,29,35,27,40)(13,23,30,19,31,38,16,32)(18,34,24,33,37,26,22,39);(1,3,4,7)(2,6,5,8)(9,12,15,28,17,29,14,27)(10,18,20,24,21,37,11,22)(13,26,30,39,31,34,16,33)(19,35,38,40,32,25,23,36);(2,5)(6,8)(9,13,15,30,17,31,14,16)(10,19,20,38,21,32,11,23)(12,26,28,39,29,34,27,33)(18,35,24,40,37,25,22,36)

Polycyclic presentation:

2^9|b^2=f,c^2=ei,d^2=f,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,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, 412378);