#10143559

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4266 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10050755);