#473097

order	512
nilpotency class	3
isoclinism	isoclinic class of #456670
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

601 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380293);