#495057

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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

710 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)(5,6,8,7)(9,10)(11,17)(12,18)(13,26)(14,30)(15,20)(16,22)(19,28)(21,34)(23,36)(24,35)(25,37)(27,31)(29,40)(32,39)(33,38);(5,7,8,6)(9,11)(10,16)(12,21)(13,27)(14,24)(15,23)(17,26)(18,31)(19,35)(20,33)(22,28)(25,38)(29,39)(30,32)(34,37)(36,40);(1,2,3,4)(9,11)(10,17)(12,24)(13,22)(14,21)(15,23)(16,26)(18,35)(19,31)(20,36)(25,39)(27,28)(29,38)(30,34)(32,37)(33,40);(9,12,15,14)(10,18,20,30)(11,21,23,24)(13,25,29,28)(16,31,33,32)(17,34,36,35)(19,26,37,40)(22,27,38,39)

Polycyclic presentation:

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

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, 402253);