Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511751 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3222 conjugacy classes of subgroups; only the 224 normal subgroups are shown.
This group has 3222 conjugacy classes of subgroups — too many to draw. Only its 224 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,14)(7,15,17,33)(8,19,21,35)(9,24,22,13)(11,26,27,16)(12,29,31,23)(18,34,25,30)(20,32,36,28);(5,7)(6,11)(8,20)(9,25)(10,17)(12,30)(13,32)(14,27)(15,19)(16,23)(18,22)(21,36)(24,28)(26,29)(31,34)(33,35);(8,21)(12,31)(15,33)(16,26)(19,35)(20,36)(23,29)(30,34);(2,4)Polycyclic presentation:
2^9|a^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,h^a=hi,e^b=eh,f^b=fi,h^b=hi,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72250058, #72250062, #72250120 |
| maximal normal subgroup of | #72250058, #72250062, #72250120 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418948);