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 | |
5097 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,28)(21,32)(22,26)(24,29)(25,31)(27,30);(1,2)(3,8)(4,7)(5,6)(10,13)(14,15)(17,19,24,31)(18,25,29,23)(20,30,22,21)(26,27,28,32);(2,6)(4,8)(9,10)(11,14)(12,13)(15,16)(17,20,24,22)(18,26,29,28)(19,30,31,21)(23,32,25,27);(2,6)(3,7)(9,11)(10,14)(12,16)(13,15)(17,21)(18,27)(19,20)(22,31)(23,26)(24,30)(25,28)(29,32);(1,3,5,7)(2,4,6,8)(17,22,24,20)(18,28,29,26)(19,30,31,21)(23,32,25,27);(1,4)(2,7)(3,6)(5,8)(17,23)(18,19)(20,27)(21,26)(22,32)(24,25)(28,30)(29,31)Polycyclic presentation:
2^9|b^2=h,c^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8302476);