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 | |
4979 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(10,13)(14,16)(17,18,22,23)(19,21,25,31)(20,27,30,24)(26,32,28,29);(1,2)(3,4)(5,6)(7,8)(11,15)(14,16)(18,23)(19,25)(24,27)(26,28);(2,6)(4,7)(9,10)(11,14)(12,13)(15,16)(18,23)(21,31)(24,27)(29,32);(2,6)(3,8)(17,19)(18,21)(20,28)(22,25)(23,31)(24,32)(26,30)(27,29);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,20,22,30)(18,24,23,27)(19,26,25,28)(21,29,31,32);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,21)(18,19)(20,29)(22,31)(23,25)(24,26)(27,28)(30,32)Polycyclic presentation:
2^9|a^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 8839759);