Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141367 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1961 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #496 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
264 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 96 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 96 points:
(1,2)(3,7,12,22)(4,14)(5,19)(6,9)(8,17)(10,16)(11,25,15,29)(13,28,26,21)(18,30)(20,31,23,27)(24,32)(33,34)(35,45,49,69)(36,51,54,41)(37,56)(38,60)(39,43)(40,48,64,57)(42,58)(44,61)(46,70)(47,71)(50,78,75,72)(52,82)(53,86,88,90)(55,68,89,81)(59,62)(63,91)(65,92,93,73)(66,84)(67,83,80,74)(76,94)(77,96)(79,87)(85,95);(1,3,6,12)(2,7)(4,15,18,11)(5,13,16,26)(8,23)(9,22)(10,21)(14,29)(17,31,32,27)(19,28)(20,24)(25,30)(33,35,46,73,61,75,59,48,39,49,58,92,38,50,37,57)(34,40,62,72,60,93,70,45,43,64,56,78,44,65,42,69)(36,52,83,76,89,47,53,87,54,85,74,91,55,77,88,84)(41,66,90,71,81,63,67,82,51,79,86,96,68,94,80,95);(1,4,16,32,6,18,5,17)(2,8,19,30,9,24,10,14)(3,11,26,31,12,15,13,27)(7,20,28,29,22,23,21,25)(33,36,38,55,39,54,61,89)(34,41,60,81,43,51,44,68)(35,47,75,85,49,77,50,52)(37,53,58,83,59,88,46,74)(40,63,65,66,64,94,93,79)(42,67,62,86,70,80,56,90)(45,71,72,95,69,96,78,82)(48,76,92,87,57,91,73,84)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=eg,d^2=ghi,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,f^c=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48564);