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 | |
4941 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(11,15)(13,16)(17,18,22,23)(19,26,25,21)(20,24,28,31)(27,32,29,30);(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,13,15,16)(18,23)(19,25)(24,31)(27,29);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(18,23)(20,28)(21,26)(27,29);(2,5)(6,8)(17,19,22,25)(18,21,23,26)(20,29,28,27)(24,32,31,30);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,28)(18,24,23,31)(19,27,25,29)(21,30,26,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,21)(18,25)(19,23)(20,30)(22,26)(24,29)(27,31)(28,32)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 8948160);