Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13432 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2044 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,13,11,16,9,15)(7,17,21,34,18,33,20,32)(8,14,24,29,23,31,22,30)(12,25,26,35,28,36,27,19);(5,7,8,19)(6,12,14,17)(9,20,22,36)(10,21,24,25)(11,18,23,35)(13,26,29,34)(15,27,30,32)(16,28,31,33);(2,4)(6,13)(7,18)(9,10)(12,26)(14,29)(15,16)(17,32)(22,24)(25,36)(27,28)(30,31)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=i,g^2=i,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,f^b=fh,h^b=hi,d^c=dh,e^c=ei,g^c=gi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72312047, #72312078, #72312079, … |
| maximal normal subgroup of | #72312047, #72312078, #72312079, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55437);