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 | |
4909 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,5)(6,8)(18,23)(19,24)(25,30)(27,31);(3,7)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,23)(19,26,24,20)(21,25,29,30)(27,32,31,28);(1,2)(3,6)(4,5)(7,8)(10,13)(11,16)(19,24)(20,26)(21,29)(25,30);(9,11,12,16)(10,14,13,15)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,31)(30,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,20)(18,24)(19,23)(21,28)(22,26)(25,31)(27,30)(29,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,21,22,29)(18,25,23,30)(19,27,24,31)(20,28,26,32)Polycyclic presentation:
2^9|b^2=g,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8969381);