Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135930 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| 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 | |
674 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)(5,6,11,16,13,17,12,30)(7,9,19,27,20,28,15,10)(8,14,24,21,25,34,31,33)(18,26,35,22,36,23,32,29);(5,7,9,16,12,19,10,17,13,20,28,30,11,15,27,6)(8,21,22,18,24,14,23,32,25,33,29,36,31,34,26,35);(2,4)(5,8)(6,14)(7,18)(9,26)(10,29)(11,24)(12,31)(13,25)(15,32)(16,21)(17,34)(19,35)(20,36)(22,27)(23,28)(30,33)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,f^b=fi,g^b=gi,d^c=dhi,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250417, #72250424, #72250437, … |
| maximal normal subgroup of | #72250417, #72250424, #72250437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55574);