#495871

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,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^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, 403067);