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 | |
4792 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,5)(6,8)(9,10,13,12)(11,15,16,14)(17,18)(19,22)(20,24)(21,25)(23,26)(27,29)(28,31)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,14,12,15);(2,5)(6,8)(17,19,23,20)(18,24,26,22)(21,27,30,28)(25,31,32,29);(2,5)(3,7)(17,20,23,19)(18,24,26,22)(21,27,30,28)(25,29,32,31);(1,2)(3,6)(4,5)(7,8)(17,21,23,30)(18,25,26,32)(19,27,20,28)(22,29,24,31);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(17,22,23,24)(18,19,26,20)(21,29,30,31)(25,27,32,28)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=h,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7995136);