#258163

order	512
nilpotency class	3
isoclinism	isoclinic class of #246079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

847 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 165359);