Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #146575 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10735 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1597 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 1597 conjugacy classes of subgroups — too many to draw. Only its 200 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:
(1,2,6,11)(3,13,17,42)(4,9,19,34)(5,21,24,38)(7,12,28,39)(8,29,32,15)(10,27,37,25)(14,43,46,63)(16,41,26,53)(18,36,49,60)(20,35,52,61)(22,31,48,58)(23,33,51,59)(30,55,57,44)(40,50,62,45)(47,56,54,64);(1,3)(2,8,12,31)(4,14)(5,22,25,15)(6,26,28,17)(7,16)(9,30,35,56)(10,13)(11,32)(18,47,23,44)(19,50,52,46)(20,45)(21,41)(24,29)(27,48)(33,43)(34,57)(36,40)(37,42,38,53)(39,58)(49,55)(51,54)(59,63,60,62)(61,64);(1,4,7,20)(2,9,12,35)(3,14,16,45)(5,23,25,18)(6,19,28,52)(8,30,31,56)(10,36,21,33)(11,34,39,61)(13,40,41,43)(15,47,22,44)(17,46,26,50)(24,51,27,49)(29,54,48,55)(32,57,58,64)(37,60,38,59)(42,62,53,63)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=di,f^d=fi,g^d=gi,g^f=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72303350, #72303355, #72303358, … |
| maximal normal subgroup of | #72303350, #72303355, #72303358, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53772);