Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511741 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14,15,18)(16,19,22,26)(17,25,23,21)(20,28,27,24);(5,7,8,12)(6,9,10,11)(14,18)(19,26)(20,27)(21,25);(14,19)(16,24)(17,23)(18,26)(20,21)(22,28)(25,27);(2,4)Polycyclic presentation:
2^9|a^2=e,b^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,h^a=hi,f^c=fh,h^c=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72286189, #72319515, #72319524, … |
| maximal normal subgroup of | #72286189, #72319515, #72319524, … |
| maximal quotient of | #72286620 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419174);