#416560

order	512
nilpotency class	3
isoclinism	isoclinic class of #397902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#392
prime radical	#60639	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1717 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)(11,20)(12,25)(13,30)(14,18)(15,19)(16,22)(17,28)(21,31)(23,33)(24,32)(26,34)(27,36)(29,35)(37,39)(38,40);(2,5)(6,7)(9,11,13,22,15,24,14,23)(10,16,18,20,19,33,30,32)(12,26,27,37,29,38,28,21)(17,34,35,31,36,40,25,39);(9,12,13,28,15,29,14,27)(10,17,18,25,19,36,30,35)(11,21,23,26,24,37,22,38)(16,31,32,40,33,39,20,34);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|b^2=eh,c^2=f,d^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^b=fi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323756);