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 | |
4858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,4)(6,7)(9,10,12,14)(11,13,15,16)(18,22)(19,25)(24,28)(27,29);(1,2,5,4)(3,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,28)(21,22)(24,30)(25,26)(27,32)(29,31);(1,2,5,4)(3,7,8,6)(17,18,21,22)(19,26,25,23)(20,24,30,28)(27,32,29,31);(1,3,5,8)(2,6,4,7)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,13,14,16)(17,19,21,25)(18,23,22,26)(20,29,30,27)(24,32,28,31);(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,29)(26,32)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=h,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7888395);