#10143550

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gi,d^2=hi,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, 10050746);