#285772

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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

576 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)(9,10,12,16,14,18,13,17)(11,19,22,24,23,20,21,15)(25,26,28,32,29,33,34,31)(27,30,36,37,35,38,40,39);(2,5)(6,8)(9,10,12,16,14,18,13,17)(11,20,22,15,23,19,21,24)(25,27,34,40,29,35,28,36)(26,30,31,39,33,38,32,37);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,21)(13,22)(14,23)(16,20)(17,19)(18,24)(26,31)(27,35)(28,34)(30,37)(32,33)(38,39);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,21,23,22)(15,20,24,19)(25,28,29,34)(26,32,33,31)(27,36,35,40)(30,37,38,39)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=e,d^2=h,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, 192968);