#329478

order	512
nilpotency class	3
isoclinism	isoclinic class of #329455
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=dh,g^b=gi,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 236674);