#327902

order	512
nilpotency class	3
isoclinism	isoclinic class of #326735
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	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

901 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)(3,8)(4,6)(5,7)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26,29,33)(27,30,34,38)(28,31,36,32)(35,37,40,39);(2,6)(3,5)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,29,34)(26,30,33,38)(28,35,36,40)(31,37,32,39);(2,7)(3,5)(6,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(26,31)(28,36)(30,37)(32,33)(35,40)(38,39);(1,3,4,5)(2,8,6,7)(12,20)(15,22)(18,24)(21,23)(27,34)(30,38)(35,40)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 235098);