#416627

order	512
nilpotency class	3
isoclinism	isoclinic class of #397950
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60639	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1740 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)(3,7)(4,5)(6,8)(9,10)(11,21)(12,26)(13,20)(14,19)(15,18)(16,23)(17,28)(22,31)(24,32)(25,33)(27,40)(29,35)(30,34)(36,37)(38,39);(2,5)(6,7)(9,11,14,24,15,25,13,23)(10,16,20,33,18,32,19,21)(12,27,30,37,29,22,28,38)(17,31,35,36,34,40,26,39);(9,12,14,28,15,29,13,30)(10,17,20,26,18,34,19,35)(11,22,24,37,25,27,23,38)(16,31,33,39,32,40,21,36);(1,3,4,8)(2,6,5,7)(10,18)(16,32)(17,34)(19,20)(21,33)(26,35)(31,40)(36,39)

Polycyclic presentation:

2^9|b^2=eh,c^2=f,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=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, 323823);