#316834

order	512
nilpotency class	3
isoclinism	isoclinic class of #262851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

664 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)(4,5)(6,8)(9,10)(11,14)(12,15)(13,17)(16,20)(18,22)(19,23)(21,24)(25,26)(27,35)(28,39)(29,34)(30,33)(31,37)(32,38)(36,40);(2,5)(6,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,21)(17,22)(20,24)(25,27,28,38,29,37,30,36)(26,31,39,40,34,35,33,32);(10,15)(11,18)(12,20)(14,21)(16,17)(22,23)(25,27,29,37)(26,32,34,40)(28,36,30,38)(31,39,35,33);(1,3,4,8)(2,6,5,7)(11,18)(14,22)(19,24)(21,23)(25,28,29,30)(26,33,34,39)(27,36,37,38)(31,40,35,32)

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=e,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, 224030);