#293018

order	512
nilpotency class	3
isoclinism	isoclinic class of #271259
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

784 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)(3,7)(4,6)(5,8)(9,10,15,11)(12,20,22,24)(13,17,14,19)(16,23,18,21)(25,26,28,31)(27,30,35,39)(29,37,36,32)(33,38,34,40);(1,3,4,8)(2,5,6,7)(9,11,15,10)(12,18,22,16)(13,17,14,19)(20,21,24,23)(25,27,28,35)(26,30,31,39)(29,33,36,34)(32,40,37,38);(9,12,13,23,15,22,14,21)(10,16,19,24,11,18,17,20)(26,31)(27,33,35,34)(30,38,39,40)(32,37);(2,6)(5,7)(9,13,15,14)(10,17,11,19)(12,21,22,23)(16,24,18,20)

Polycyclic presentation:

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

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, 200214);