#286455

order	512
nilpotency class	3
isoclinism	isoclinic class of #281787
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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

596 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)(9,10,14,16,12,17,13,18)(11,19,20,23,21,15,22,24)(25,26,30,32,29,33,28,34)(27,31,37,40,35,39,36,38);(2,5)(6,8)(9,11,13,22,12,21,14,20)(10,15,18,23,17,19,16,24)(25,27,30,37,29,35,28,36)(26,31,32,40,33,39,34,38);(1,3,4,7)(2,6,5,8)(10,16)(11,20)(13,14)(15,19)(17,18)(21,22)(26,32)(27,35)(28,30)(31,38)(33,34)(39,40);(2,5)(6,8)(11,21)(15,19)(20,22)(23,24)(26,33)(27,35)(32,34)(36,37)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 193651);