Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141343 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1957 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #492 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
297 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 297 conjugacy classes of subgroups — too many to draw. Only its 98 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)(3,7,12,21)(4,14)(5,17)(6,9)(8,15)(10,18)(11,23,24,32)(13,27,26,20)(16,28)(19,30,31,25)(22,29)(33,34,37,42,39,43,38,54)(35,44,46,61,47,60,53,62)(36,48,50,56,52,64,51,41)(40,55,57,45,59,63,58,49);(1,3,6,12)(2,7)(4,11,16,24)(5,13,18,26)(8,19)(9,21)(10,20)(14,23)(15,25,29,30)(17,27)(22,31)(28,32)(33,35,36,49,38,46,51,55,39,47,52,45,37,53,50,63)(34,40,56,44,42,58,64,62,43,59,41,60,54,57,48,61);(1,4,5,15,6,16,18,29)(2,8,10,14,9,22,17,28)(3,11,13,25,12,24,26,30)(7,19,20,23,21,31,27,32)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=e,d^2=ghi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dh,f^b=fh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48557);