#504475

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	#25434	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

602 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 411671);