#494861

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

802 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,h^a=hi,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402057);