Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131671 |
| 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 | #62383 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
513 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 513 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)(33,34,39,46,40,47,64,78,41,48,63,77,38,45,37,44)(35,49,53,71,54,75,51,68,55,82,61,42,52,69,89,70)(36,56,59,76,58,74,62,80,60,79,66,72,65,43,57,73)(50,83,86,94,85,93,84,95,87,96,91,92,90,81,88,67);(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)(36,50)(37,61)(38,54)(39,51)(40,52)(41,55)(43,67)(44,75)(45,70)(46,49)(47,68)(48,71)(53,64)(56,92)(57,91)(58,90)(59,84)(60,87)(62,86)(63,89)(65,85)(66,88)(69,78)(72,96)(73,93)(74,95)(76,83)(77,82)(79,94)(80,81);(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)(35,50)(37,62)(38,65)(39,66)(40,58)(41,60)(42,67)(44,76)(45,79)(46,80)(47,56)(48,74)(49,81)(51,88)(52,90)(53,91)(54,85)(55,87)(57,64)(59,63)(61,86)(68,92)(69,96)(70,94)(71,95)(72,78)(73,77)(75,83)(82,93)(84,89)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,d^b=dh,e^b=ei,f^b=fgh,h^b=hi,d^c=di,e^c=egi,f^c=fgi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38971);