Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511785 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #23908 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2346 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 2346 conjugacy classes of subgroups — too many to draw. Only its 188 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)(7,17)(8,13)(9,14)(10,15)(11,16)(12,19)(18,29)(20,27)(21,26)(22,30)(23,31)(24,32)(25,33)(28,34)(35,36);(5,7,11,8)(6,12,16,26)(9,20,23,18)(10,19,24,21)(13,15,17,32)(14,28,31,25)(22,34,35,33)(27,36,29,30);(5,8,11,7)(6,13,16,17)(9,18,23,20)(10,19,24,21)(12,15,26,32)(14,29,31,27)(22,34,35,33)(25,36,28,30);(2,4)(5,9,11,23)(6,14,16,31)(7,18,8,20)(10,22,24,35)(12,25,26,28)(13,27,17,29)(15,30,32,36)(19,33,21,34)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,d^b=di,f^b=fhi,d^c=di,e^c=eh,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72279992, #72280213, #72306634, … |
| maximal normal subgroup of | #72279992, #72280213, #72306634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418985);