#411235

order	512
nilpotency class	3
isoclinism	isoclinic class of #399126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
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

717 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)(3,6)(4,5)(7,8)(9,10,16,23,14,20,15,21)(11,17,24,32,25,13,19,30)(12,18,31,38,28,36,29,37)(22,33,26,34,39,27,35,40);(9,11,16,24,14,25,15,19)(10,17,21,30,20,13,23,32)(12,26,31,39,28,35,29,22)(18,34,37,33,36,40,38,27);(2,5)(6,8)(9,12,15,29,14,28,16,31)(10,18,21,37,20,36,23,38)(11,22,24,26,25,39,19,35)(13,27,30,40,17,33,32,34);(1,3,4,7)(2,6,5,8)(9,13,14,17)(10,19,20,24)(11,23,25,21)(12,27,28,33)(15,30,16,32)(18,35,36,26)(22,38,39,37)(29,40,31,34)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318431);