#493237

order	512
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

740 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gi,d^2=fi,e^2=hi,h^2=i,b^a=be,e^a=eh,h^a=hi,e^b=eh,h^b=hi,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400433);