#466842

order	512
nilpotency class	3
isoclinism	isoclinic class of #457102
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23426	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1106 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=efi,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,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, 374038);