Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13310 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1999 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(7,17)(8,22)(9,25)(10,24)(12,30)(13,14)(15,28)(18,34)(19,27)(20,21)(23,29)(26,33)(31,36)(32,35);(5,6)(7,18,19,35)(8,14)(9,26)(10,28)(11,16)(12,20,31,23)(13,22)(15,24)(17,32,27,34)(21,30,29,36)(25,33);(5,7)(6,12)(8,23)(9,27)(10,29)(11,19)(13,18)(14,30)(15,34)(16,31)(17,22)(20,25)(21,24)(26,35)(28,32)(33,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fh,h^b=hi,d^c=di,e^c=eh,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249930, #72249936, #72249977, … |
| maximal normal subgroup of | #72249930, #72249936, #72249977, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55306);