#259005

order	512
nilpotency class	3
isoclinism	isoclinic class of #246415
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×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

820 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)(11,18)(12,16)(13,15)(14,19)(17,21)(20,24)(22,23)(25,26)(27,31)(28,30)(29,32)(33,40)(34,37)(35,38)(36,39);(2,5)(6,7)(9,11,12,19)(10,14,16,18)(13,22,21,20)(15,24,17,23)(26,30)(27,33)(29,35)(31,39)(34,36)(37,40);(10,15)(13,21)(14,23)(16,17)(18,24)(20,22)(25,27,28,34)(26,31,30,37)(29,36,35,33)(32,39,38,40);(1,3,4,8)(2,6,5,7)(10,16)(14,18)(15,17)(23,24)(26,30)(31,37)(32,38)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 166201);