#493236

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

778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gi,d^2=f,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, 400432);