#465359

order	512
nilpotency class	3
isoclinism	isoclinic class of #455116
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

911 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=egi,c^2=ef,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,g^c=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, 372555);