#256784

order	512
nilpotency class	3
isoclinism	isoclinic class of #246847
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

720 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)(3,8)(4,6)(5,7)(9,10)(11,19)(12,21)(13,17)(14,20)(15,18)(16,22)(23,24)(25,26)(27,33)(28,30)(29,37)(31,35)(32,36)(34,40)(38,39);(1,3,5,4)(2,6,7,8)(9,11)(10,12)(13,21)(14,16)(15,24)(17,19)(18,22)(20,23)(26,30)(29,36)(33,40)(35,38);(1,4,5,3)(2,6,7,8)(9,12,13,19)(10,16,17,24)(11,20,21,18)(14,23,15,22)(25,27)(26,31)(28,34)(29,38)(30,39)(32,40)(33,37)(35,36);(2,7)(6,8)(10,17)(12,19)(18,20)(22,23)(26,30)(27,34)(32,37)(35,38)

Polycyclic presentation:

2^9|b^2=eh,c^2=ei,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dh,f^c=fh,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 163980);