#450934

order	512
nilpotency class	3
isoclinism	isoclinic class of #417566
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1149 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358130);