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 | |
5018 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)(19,26)(20,24)(21,23)(22,25)(27,32)(28,30)(29,31);(1,2)(3,7)(4,5)(6,8)(9,10,12,13)(11,14,15,16)(17,19)(18,20)(21,26)(22,30)(23,24)(25,32)(27,31)(28,29);(2,5)(6,7)(18,21)(19,24)(25,29)(27,30);(1,2)(3,6)(4,5)(7,8)(17,20,23,26)(18,24,21,19)(22,28,31,32)(25,30,29,27);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,14,13,16)(17,21,23,18)(19,26,24,20)(22,29,31,25)(27,32,30,28);(1,3)(2,6)(4,8)(5,7)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,30)(26,32)Polycyclic presentation:
2^9|b^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7640568);