#292148

order	512
nilpotency class	3
isoclinism	isoclinic class of #273051
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	#2847	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

947 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)(11,14)(12,15)(13,16)(17,24)(18,22)(19,21)(20,23)(26,31)(27,35)(28,30)(32,36)(37,38)(39,40);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,19)(13,18)(15,21)(16,22)(17,20)(23,24)(25,26)(27,36)(28,33)(29,31)(30,34)(32,38)(35,39)(37,40);(9,12,13,20)(10,15,16,23)(11,17)(14,21)(18,19)(22,24)(25,27,30,38,29,37,28,35)(26,32,33,36,31,39,34,40);(2,6)(5,7)(12,20)(15,23)(17,19)(21,24)(25,28,29,30)(26,33,31,34)(27,35,37,38)(32,36,39,40)

Polycyclic presentation:

2^9|b^2=eh,c^2=fi,d^2=h,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,g^c=ghi

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, 199344);