Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8674 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #380 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1797 |
| prime radical | C2×#41 | C4⋊C4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
279 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 279 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 256 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,11)(4,15,17,8)(5,18)(6,10)(7,13)(9,19)(12,25,27,23)(14,24)(16,22,31,30)(20,26,29,32)(21,28)(33,34,37,43,39,42,52,58,38,44,51,59,36,40,50,53)(35,45,48,60,47,41,54,56,49,55,62,63,46,57,61,64);(1,3,5,13,6,14,19,28)(2,7,9,11,10,21,18,24)(4,12,16,26,17,27,31,32)(8,20,22,23,15,29,30,25)(34,40,44,42)(41,45,57,55)(43,53,59,58)(56,60,64,63);(1,4)(2,8)(3,12)(5,16)(6,17)(7,20)(9,22)(10,15)(11,23)(13,26)(14,27)(18,30)(19,31)(21,29)(24,25)(28,32)(33,35)(34,41)(36,46)(37,48)(38,49)(39,47)(40,45)(42,55)(43,56)(44,57)(50,61)(51,62)(52,54)(53,60)(58,63)(59,64)Polycyclic presentation:
2^9|a^2=f,b^2=dh,d^2=g,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,e^c=ei,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41909);