#316381

order	512
nilpotency class	3
isoclinism	isoclinic class of #262851
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

668 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)(11,17)(12,19)(14,21)(16,23)(18,20)(22,24)(25,26)(27,32)(28,39)(29,34)(30,35)(31,36)(33,37)(38,40);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,16)(13,15)(17,21)(18,22)(19,23)(20,24)(25,27,30,37,31,28,29,38)(26,32,35,33,36,39,34,40);(9,11,13,18)(10,14,15,22)(12,20,19,17)(16,24,23,21)(25,28)(26,33)(27,31)(29,38)(30,37)(32,35)(34,39)(36,40);(2,6)(5,7)(10,15)(14,22)(16,23)(21,24)(25,29,31,30)(26,34,36,35)(27,37,28,38)(32,33,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 223577);