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 | |
1226 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 1226 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(7,10)(8,9)(11,12)(13,14,17,21)(15,18,24,27)(16,25,23,20)(19,28,26,22);(6,9)(10,12);(2,4)(5,7,8,11)(6,10,9,12)(14,18)(15,22)(16,23)(19,20)(21,27)(24,28)(25,26);(14,18)(15,22)(16,23)(19,20)(21,27)(24,28)(25,26)Polycyclic presentation:
2^9|a^2=i,c^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gh,h^a=hi,f^c=fh,g^c=gh,h^c=hi,f^d=fh,g^d=gh,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72286189, #72317198, #72317317, … |
| maximal normal subgroup of | #72286189, #72317198, #72317317, … |
| maximal quotient of | #72286620 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419277);