#505183

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	#33524	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

648 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)(11,22)(12,14)(13,29)(15,19)(16,20)(17,21)(18,25)(23,40)(24,39)(26,27)(28,31)(30,37)(32,35)(33,36)(34,38);(2,5)(6,8)(9,11,14,25,17,26,16,33)(10,18,20,22,21,36,12,27)(13,24,15,32,31,38,30,40)(19,34,29,35,37,39,28,23);(1,3,4,7)(2,6,5,8)(9,12,16,21,17,20,14,10)(11,23,33,39,26,35,25,34)(13,28,30,37,31,29,15,19)(18,32,27,24,36,40,22,38);(2,5)(6,8)(9,13,16,30,17,31,14,15)(10,19,12,28,21,37,20,29)(11,24,33,40,26,38,25,32)(18,34,27,23,36,39,22,35)

Polycyclic presentation:

2^9|b^2=f,c^2=ei,d^2=fi,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, 412379);