#252516

order	512
nilpotency class	3
isoclinism	isoclinic class of #246159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

864 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)(4,6)(5,8)(9,10)(11,12)(13,17)(14,19)(15,20)(16,22)(18,23)(21,24)(26,31)(27,35)(28,30)(32,36)(37,38)(39,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,17)(13,20)(14,21)(16,23)(18,24)(19,22)(25,26)(27,36)(28,33)(29,31)(30,34)(32,37)(35,39)(38,40);(9,12)(10,16)(11,19)(13,15)(14,24)(17,21)(18,20)(22,23)(25,27,29,38)(26,32,31,39)(28,37,30,35)(33,40,34,36);(2,6)(5,7)(10,17)(12,15)(18,19)(22,24)(25,28,29,30)(26,33,31,34)(27,37,38,35)(32,40,39,36)

Polycyclic presentation:

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

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, 159712);