Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #54968 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4161 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,14)(12,13)(15,18)(16,22)(17,23)(19,24)(20,21)(25,26,32,31)(27,36,34,29)(28,33,38,37)(30,35,39,40);(2,3)(6,7)(9,11,15,17)(10,14,18,23)(12,20,22,19)(13,21,16,24)(25,27)(26,29)(28,35)(30,33)(31,36)(32,34)(37,39)(38,40);(1,2,5,3)(4,6,8,7)(9,12,15,22)(10,16,18,13)(11,19,17,20)(14,21,23,24)(25,28,32,38)(26,33,31,37)(27,35,34,40)(29,30,36,39);(9,12)(10,13)(11,20)(14,21)(15,22)(16,18)(17,19)(23,24)(25,29,32,36)(26,34,31,27)(28,30,38,39)(33,40,37,35);(1,3,5,2)(4,6,8,7)(9,13)(10,12)(11,21)(14,20)(15,16)(17,24)(18,22)(19,23)(25,30)(26,35)(27,37)(28,36)(29,38)(31,40)(32,39)(33,34);(1,4)(2,6)(3,7)(5,8)(9,14)(10,17)(11,18)(12,21)(13,19)(15,23)(16,20)(22,24)(25,31,32,26)(27,36,34,29)(28,37,38,33)(30,35,39,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gi,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9755696);