#495825

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, prime radical	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=i,d^2=gi,f^2=hi,h^2=i,b^a=be,c^a=cf,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, 403021);