Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131664 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #466 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
506 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 506 conjugacy classes of subgroups — too many to draw. Only its 90 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:
(1,2,6,12,7,10,5,11)(3,13,15,26,16,8,21,23)(4,17,19,9,20,25,18,24)(14,28,29,22,30,32,31,27)(33,34,39,46,40,45,61,76,41,47,60,77,38,62,37,44)(35,48,52,69,51,75,50,66,53,79,59,42,64,67,86,68)(36,54,57,71,56,43,70,72,58,73,90,94,63,74,55,88)(49,80,83,93,84,96,81,65,85,91,89,87,82,78,95,92);(1,3)(2,8)(4,14)(5,15)(6,21)(7,16)(9,22)(10,13)(11,23)(12,26)(17,32)(18,29)(19,31)(20,30)(24,27)(25,28)(33,35)(34,42)(36,49)(37,59)(38,51)(39,50)(40,64)(41,53)(43,65)(44,75)(45,66)(46,48)(47,69)(52,61)(54,87)(55,89)(56,82)(57,81)(58,85)(60,86)(62,68)(63,84)(67,76)(70,83)(71,80)(72,78)(73,93)(74,92)(77,79)(88,96)(90,95)(91,94);(1,4,7,20)(2,9,10,24)(3,14,16,30)(5,19,6,18)(8,22,13,27)(11,25,12,17)(15,31,21,29)(23,28,26,32)(33,36)(34,43)(35,49)(37,55)(38,63)(39,57)(40,56)(41,58)(42,65)(44,71)(45,73)(46,72)(47,74)(48,78)(50,81)(51,84)(52,83)(53,85)(54,62)(59,89)(60,90)(61,70)(64,82)(66,93)(67,91)(68,87)(69,92)(75,80)(76,94)(77,88)(79,96)(86,95)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=hi,e^2=gi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,f^b=fgh,h^b=hi,e^c=egi,f^c=fg53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38962);