Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13459 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #498 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2084 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,9,14,11,15,10,25)(7,16,17,33,19,32,20,31)(8,21,23,30,22,36,24,13)(12,26,27,18,28,34,29,35);(5,7,8,18,10,17,24,34,11,19,22,35,9,20,23,26)(6,12,13,16,14,29,21,31,15,28,30,32,25,27,36,33);(1,2,3,4)(7,17,19,20)(8,22)(12,27,28,29)(13,30)(16,31,32,33)(18,26,35,34)(21,36)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,e^b=ei,g^b=gi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250502, #72250514, #72250540, … |
| maximal normal subgroup of | #72250502, #72250514, #72250540, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56388);