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 | |
5118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,7)(6,8)(10,13)(14,16)(17,18,23,24)(19,21,26,31)(20,27,22,25)(28,32,29,30);(2,5)(6,8)(10,13)(11,15)(18,24)(21,31)(25,27)(30,32);(1,2)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(17,19,23,26)(18,21,24,31)(20,28,22,29)(25,30,27,32);(9,11)(10,14)(12,15)(13,16)(17,20,23,22)(18,25,24,27)(19,28,26,29)(21,30,31,32);(1,2)(3,6)(4,5)(7,8)(17,21)(18,26)(19,24)(20,30)(22,32)(23,31)(25,29)(27,28);(1,3)(2,6)(4,7)(5,8)(17,22,23,20)(18,27,24,25)(19,29,26,28)(21,32,31,30)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000327);