Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #327 |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | C2×#405 | C22 |
| socle | C2 | #7028 |
| prime radical | C24 | #104 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×#405 ▷ C22×C4 ▷ C2 ▷ C1 | |
310 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 82 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 — 2 generators on 64 points:
(1,2,5,11,7,12,27,39,8,13,28,40,6,29,26,54)(3,14,16,41,17,36,46,60,19,42,48,62,18,43,47,63)(4,20,22,49,25,38,53,61,23,50,52,64,24,10,35,37)(9,30,31,45,33,56,58,44,34,55,59,21,32,51,57,15);(1,3)(2,9)(4,21,23,45)(5,16)(6,18)(7,17)(8,19)(10,36,38,43)(11,31)(12,33)(13,34)(14,20,42,50)(15,24,44,25)(22,51,52,56)(26,47)(27,46)(28,48)(29,32)(30,35,55,53)(37,60,61,63)(39,58)(40,59)(41,49,62,64)(54,57)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,g^a=gi,c^b=cg,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 613);