Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190022 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4198 conjugacy classes of subgroups; only the 438 normal subgroups are shown.
This group has 4198 conjugacy classes of subgroups — too many to draw. Only its 438 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,8,12,9,10,16,17)(7,13,15,20,14,11,18,19);(6,10)(8,16)(11,13)(15,18);(5,7)(6,11,10,13)(8,15)(9,14)(12,19,17,20)(16,18);(2,4)Polycyclic presentation:
2^9|a^2=h,c^2=e,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^b=hi,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72219299, #72219304, #72219305, … |
| maximal normal subgroup of | #72219299, #72219304, #72219305, … |
| maximal quotient of | #72238525 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299429);