Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437798 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3585 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,11,12,10,16,9,15)(7,17,23,28,20,13,18,25)(8,14,24,26,22,30,21,29)(19,31,34,36,33,27,32,35);(6,12)(7,18)(8,21)(9,11)(15,16)(20,23)(22,24)(25,28)(26,29)(27,35)(31,36)(32,34);(2,4)(5,7)(6,13)(8,19)(9,18)(10,20)(11,23)(12,25)(14,27)(15,28)(16,17)(21,32)(22,33)(24,34)(26,35)(29,36)(30,31);(5,8)(6,14)(7,19)(9,21)(10,22)(11,24)(12,26)(13,27)(15,29)(16,30)(17,31)(18,32)(20,33)(23,34)(25,35)(28,36);(7,20)(13,17)(18,23)(19,33)(25,28)(27,31)(32,34)(35,36)Polycyclic presentation:
2^9|a^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72316840 |
| maximal normal subgroup of | #72316840 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344995);