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 | |
4893 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,12,14)(11,13,15,16)(17,18)(19,20)(21,25)(22,23)(24,26)(27,28)(29,32)(30,31);(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16);(1,3)(2,5)(4,8)(6,7)(18,23)(19,26)(25,31)(27,32);(1,2)(3,6)(4,7)(5,8)(17,19)(18,24)(20,23)(21,27)(22,26)(25,29)(28,31)(30,32);(1,2)(3,5)(4,7)(6,8)(17,20,22,24)(18,19,23,26)(21,29,30,28)(25,32,31,27);(1,3)(2,6)(4,8)(5,7)(9,11,12,15)(10,13,14,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,29)(26,32)Polycyclic presentation:
2^9|a^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7566272);