#457027

order	512
nilpotency class	3
isoclinism	isoclinic class of #456417
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1122 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)(3,6)(4,5)(7,8)(9,10,14,19,16,20,15,30)(11,17,22,33,23,34,24,39)(12,25,27,35,13,28,29,18)(21,36,37,32,26,40,38,31);(2,5)(6,8)(9,11,14,22,16,23,15,24)(10,17,19,33,20,34,30,39)(12,26,27,38,13,21,29,37)(18,32,25,40,35,31,28,36);(3,7)(6,8)(9,12,16,13)(10,18,20,35)(11,21,23,26)(14,29,15,27)(17,31,34,32)(19,28,30,25)(22,38,24,37)(33,40,39,36);(1,3,4,7)(2,6,5,8)(9,13,16,12)(10,18,20,35)(11,21,23,26)(14,29,15,27)(17,32,34,31)(19,25,30,28)(22,37,24,38)(33,40,39,36)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,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, 364223);