#258127

order	512
nilpotency class	3
isoclinism	isoclinic class of #246159
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	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

977 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)(11,17)(12,15)(13,16)(14,19)(18,21)(20,23)(22,24)(28,36)(30,39)(32,37)(35,38);(2,5)(6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,24)(21,23)(25,26,29,31)(27,33,34,40)(28,37,36,32)(30,38,39,35);(9,12)(10,15)(11,18)(13,20)(14,21)(16,23)(17,22)(19,24)(25,27,29,34)(26,30,31,39)(28,38,36,35)(32,40,37,33);(1,3,4,8)(2,6,5,7)(10,16)(12,20)(14,17)(18,24)(27,34)(30,39)(33,40)(35,38)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=ehi,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=dh,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, 165323);