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 | |
4866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(10,13)(14,16)(17,18,22,23)(19,21,26,24)(20,28,29,25)(27,31,32,30);(1,2)(3,4)(5,6)(7,8)(11,15)(14,16)(18,23)(19,26)(20,29)(30,31);(2,6)(4,7)(9,10)(11,14)(12,13)(15,16)(18,23)(21,24)(25,28)(30,31);(9,11)(10,14)(12,15)(13,16)(17,19,22,26)(18,24,23,21)(20,27,29,32)(25,31,28,30);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,25)(19,27)(21,30)(22,29)(23,28)(24,31)(26,32);(1,4,5,7)(2,3,6,8)(17,21,22,24)(18,26,23,19)(20,30,29,31)(25,32,28,27)Polycyclic presentation:
2^9|a^2=g,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8137930);