#256288

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
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

877 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)(11,19)(12,18)(13,17)(14,16)(15,22)(20,24)(21,23)(25,26,30,29)(27,31,34,37)(28,32,36,38)(33,39,40,35);(2,5)(6,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,23)(17,19)(18,24)(25,27,30,34)(26,31,29,37)(28,35,36,39)(32,33,38,40);(1,3,4,7)(2,6,5,8)(10,16)(11,20)(12,14)(17,18)(21,22)(23,24)(25,28)(26,32)(27,33)(29,38)(30,36)(31,39)(34,40)(35,37);(2,5)(6,8)(9,12,13,14)(10,16,17,18)(11,21,22,20)(15,23,19,24)(25,26,30,29)(27,31,34,37)(28,32,36,38)(33,39,40,35)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 163484);