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 | |
6976 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)(8,9)(10,11)(12,15)(13,18)(16,20)(17,19)(24,32)(28,33)(30,36)(34,35);(1,2,3,4)(7,15)(12,14)(16,19)(17,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(6,11)(8,13)(12,14)(16,19)(22,26)(27,29)(28,33)(34,35);(5,7)(6,12)(8,16)(9,17)(10,15)(11,14)(13,19)(18,20)(21,23,25,31)(22,27,26,29)(24,30,32,36)(28,34,33,35);(5,8,10,13)(6,9,11,18)(7,16,15,19)(12,17,14,20)(21,22)(23,27)(24,28)(25,26)(29,31)(30,34)(32,33)(35,36);(5,9)(6,13)(7,17)(8,11)(10,18)(12,19)(14,16)(15,20)(21,24)(22,28)(23,30)(25,32)(26,33)(27,34)(29,35)(31,36)Polycyclic presentation:
2^9|b^2=g,d^2=i,e^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, 7551099);