Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 | |
6538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(2,4)(5,6)(7,14)(8,11)(9,17)(10,13)(12,18)(15,19)(16,20)(22,26)(24,32)(27,29)(30,36);(1,2,3,4)(5,7,10,8)(6,11,13,14)(9,15,18,16)(12,19,17,20);(7,8)(11,14)(15,16)(19,20)(21,22,25,26)(23,29,31,27)(24,28,32,33)(30,35,36,34);(5,6)(7,14)(8,11)(9,12)(10,13)(15,20)(16,19)(17,18)(21,23,25,31)(22,27,26,29)(24,30,32,36)(28,34,33,35);(5,8,10,7)(6,11,13,14)(9,16,18,15)(12,19,17,20)(21,24)(22,28)(23,30)(25,32)(26,33)(27,34)(29,35)(31,36);(5,9)(6,12)(7,15)(8,16)(10,18)(11,19)(13,17)(14,20)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=h,e^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7621779);