#285786

order	512
nilpotency class	3
isoclinism	isoclinic class of #281659
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

618 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)(3,6)(4,5)(7,8)(9,10,14,16,13,17,12,23)(11,18,22,19,21,15,20,24)(25,26)(27,34)(28,38)(29,33)(30,32)(31,37)(35,40)(36,39);(2,5)(6,8)(9,11,14,22,13,21,12,20)(10,15,16,24,17,18,23,19)(25,27,29,37)(26,31,33,34)(28,36,30,35)(32,39,38,40);(1,3,4,7)(2,6,5,8)(9,10)(11,19)(12,16)(13,17)(14,23)(15,20)(18,22)(21,24)(27,35)(28,30)(31,39)(32,38)(34,40)(36,37);(2,5)(6,8)(9,12,13,14)(10,16,17,23)(11,20,21,22)(15,24,18,19)(25,28,29,30)(26,32,33,38)(27,36,37,35)(31,39,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192982);