#504382

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	#3061	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

525 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)(12,15)(13,26)(14,24)(16,20)(17,25)(18,31)(19,36)(21,34)(23,29)(27,33)(28,39)(30,40)(32,37)(35,38);(9,11,15,26)(10,17,12,28)(13,24,22,38)(14,29,35,18)(16,37,34,27)(19,32,30,33)(20,31,21,23)(25,40,39,36);(2,5)(6,8)(9,12,15,10)(11,23,26,31)(13,29,22,18)(14,30,35,19)(16,20,34,21)(17,37,28,27)(24,36,38,40)(25,33,39,32);(1,3,4,7)(2,6,5,8)(9,13,15,22)(10,18,12,29)(11,24,26,38)(14,28,35,17)(16,32,34,33)(19,37,30,27)(20,25,21,39)(23,36,31,40)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=f,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,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, 411578);