#495878

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
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

676 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)(3,4)(5,6,8,7)(9,10)(11,20)(12,17)(13,28)(14,27)(15,19)(16,24)(18,29)(21,34)(22,35)(23,36)(25,38)(26,37)(30,31)(32,40)(33,39);(9,11,21,39,14,23,13,24,15,25,29,40,12,26,30,35)(10,16,31,36,17,33,18,20,19,22,28,37,27,32,34,38);(2,4)(9,10)(11,22)(12,27)(13,18)(14,17)(15,19)(16,25)(20,35)(21,31)(23,32)(24,38)(26,33)(28,29)(30,34)(36,40)(37,39);(5,7,8,6)(9,12,15,14)(10,17,19,27)(11,23,25,26)(13,21,30,29)(16,32,22,33)(18,28,34,31)(20,36,38,37)(24,40,35,39)

Polycyclic presentation:

2^9|a^2=g,b^2=ehi,d^2=gi,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,c^b=ci,d^b=di,d^c=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, 403074);