#329910

order	512
nilpotency class	3
isoclinism	isoclinic class of #329515
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

562 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 237106);