#495122

order	512
nilpotency class	4
isoclinism	isoclinic class of #33605
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1346 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402318);