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 | |
6498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(2,4)(5,6)(7,13)(8,16)(9,10)(11,14)(12,17)(15,18)(19,20)(22,26)(23,29)(28,33)(30,35);(1,2)(3,4)(7,14)(11,13)(15,20)(18,19)(21,22,25,26)(23,27,29,32)(24,28,31,33)(30,34,35,36);(6,10)(11,13)(12,16)(18,19)(21,22)(23,27)(24,28)(25,26)(29,32)(30,34)(31,33)(35,36);(5,7)(6,11)(8,15)(9,14)(10,13)(12,18)(16,19)(17,20)(21,23,25,29)(22,27,26,32)(24,30,31,35)(28,34,33,36);(5,6)(7,11)(8,12)(9,10)(13,14)(15,18)(16,17)(19,20);(5,8,9,17)(6,12,10,16)(7,15,14,20)(11,18,13,19)(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36)Polycyclic presentation:
2^9|b^2=i,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7551234);