#286426

order	512
nilpotency class	3
isoclinism	isoclinic class of #281787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

611 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)(9,10,14,19,13,18,12,17)(11,20,21,24,22,15,23,16)(25,26,28,33,29,32,34,31)(27,30,36,37,35,38,40,39);(2,5)(6,8)(9,11,14,21,13,22,12,23)(10,15,19,16,18,20,17,24)(25,27,34,40,29,35,28,36)(26,30,31,39,32,38,33,37);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,21)(13,22)(14,23)(15,19)(17,20)(18,24)(26,31)(27,35)(28,34)(30,37)(32,33)(38,39);(2,5)(6,8)(9,12,13,14)(10,17,18,19)(11,21,22,23)(15,16,20,24)(26,32)(27,35)(31,33)(36,40)

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=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, 193622);