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 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5090 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)(3,4)(5,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18,23,26)(19,24,20,21)(22,25,30,32)(27,31,28,29);(9,11,12,16)(10,13,14,15)(17,19,23,20)(18,21,26,24)(22,28,30,27)(25,31,32,29);(2,6)(3,7)(19,20)(21,24)(27,28)(29,31);(1,3,5,7)(2,4,6,8)(17,20,23,19)(18,24,26,21)(22,28,30,27)(25,31,32,29);(1,4)(2,7)(3,6)(5,8)(17,21,23,24)(18,19,26,20)(22,29,30,31)(25,27,32,28);(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=hi,e^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72318126, #72318169, #72318486, … |
| maximal normal subgroup of | #72318126, #72318169, #72318486, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7650394);