#316945

order	512
nilpotency class	3
isoclinism	isoclinic class of #262883
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

566 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cg,d^b=dh,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, 224141);