Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152839 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13149 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
209 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 209 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,7,14,9,17,10,16)(3,18,21,5,13,32,23,33)(4,20,27,24,30,49,29,54)(6,12,37,47,38,48,19,22)(8,40,39,15,36,46,31,34)(11,41,43,26,45,55,50,35)(25,44,51,61,60,52,59,56)(28,57,58,64,62,42,63,53);(1,3,19,40,8,22,33,2,10,21,38,15,31,12,32,14,9,13,37,46,36,47,5,17,7,23,6,34,39,48,18,16)(4,24,55,44,28,57,25,50,27,20,35,56,58,53,51,45,30,54,41,52,62,42,60,43,29,49,26,61,63,64,59,11);(1,4,5,25,31,58,19,41,7,29,32,59,8,28,6,35,9,30,33,60,39,63,37,55,10,27,18,51,36,62,38,26)(2,11,12,42,46,56,23,24,16,43,22,53,15,44,13,49,17,45,48,57,40,61,21,54,14,50,47,64,34,52,3,20)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=d,d^2=ghi,e^2=gi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72225559, #72225568, #72225574, … |
| maximal normal subgroup of | #72225559, #72225568, #72225574, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60037);