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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,28)(19,26,21,24)(20,25,27,22)(29,30,31,32);(2,3)(6,7)(10,13)(14,15)(19,21)(24,26)(29,31)(30,32);(9,10)(11,14)(12,13)(15,16)(17,19,23,21)(18,24,28,26)(20,29,27,31)(22,30,25,32);(2,3)(4,8)(9,11,12,16)(10,14,13,15)(17,20)(18,25)(19,29)(21,31)(22,28)(23,27)(24,32)(26,30);(1,3,5,2)(4,7,8,6)(17,21,23,19)(18,26,28,24)(20,31,27,29)(22,32,25,30);(1,4)(2,6)(3,7)(5,8)(17,22,23,25)(18,27,28,20)(19,30,21,32)(24,31,26,29)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=hi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10349753);