Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134415 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1804 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,15,10,14,25,34,11,17,26,33,8,16,24,13)(7,18,22,35,21,31,36,28,23,30,20,32,19,12,27,29);(1,2)(3,4)(5,7,8,19,11,23,10,21)(6,12,14,18,17,31,16,30)(9,22,24,27,26,20,25,36)(13,32,15,29,34,35,33,28);(2,4)(6,13)(7,19)(8,10)(9,24)(12,28)(14,33)(15,16)(17,34)(18,35)(20,22)(21,23)(25,26)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,e^2=hi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,e^b=ei,d^c=di,e^c=eh,f^c=fi,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250156, #72250178, #72250225, … |
| maximal normal subgroup of | #72250156, #72250178, #72250225, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55422);