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 | |
6937 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)(7,14)(8,16)(9,12)(10,13)(11,18)(15,17)(19,20);(1,2)(3,4)(22,26)(27,33)(28,34)(35,36);(5,7,10,12)(6,9,13,14)(8,15,18,19)(11,17,16,20)(21,22,25,26)(23,27,30,33)(24,28,32,34)(29,35,31,36);(5,8)(6,11)(7,15)(9,17)(10,18)(12,19)(13,16)(14,20);(5,6)(7,9)(8,11)(10,13)(12,14)(15,17)(16,18)(19,20)(21,23,25,30)(22,27,26,33)(24,31,32,29)(28,36,34,35);(5,9)(6,12)(7,13)(8,17)(10,14)(11,19)(15,16)(18,20)(21,24,25,32)(22,28,26,34)(23,29,30,31)(27,35,33,36)Polycyclic presentation:
2^9|c^2=hi,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7561793);