#417130

order	512
nilpotency class	3
isoclinism	isoclinic class of #397902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1383 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)(12,17)(13,28)(14,18)(15,19)(16,22)(21,37)(23,36)(24,31)(25,30)(26,34)(27,35)(29,33)(32,38)(39,40);(2,5)(6,7)(9,11,13,22,15,23,14,24)(10,16,18,20,19,31,28,36)(12,25,26,38,27,39,29,21)(17,32,33,30,35,37,34,40);(9,12,14,26,15,27,13,29)(10,17,18,34,19,35,28,33)(11,21,22,39,23,38,24,25)(16,30,36,37,31,40,20,32);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|b^2=eh,c^2=fg,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 324326);