Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8479 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1779 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
509 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 509 conjugacy classes of subgroups — too many to draw. Only its 110 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)(3,11)(4,15,17,8)(5,18)(6,10)(7,13)(9,20)(12,26,28,25)(14,24)(16,23,32,30)(19,22)(21,27,29,31)(33,34,37,42,38,43,52,62,39,44,53,63,54,64,75,85)(35,40,47,57,48,58,71,81,46,56,69,79,70,80,90,94)(36,41,50,60,51,61,74,84,49,59,72,82,73,83,91,95)(45,65,67,86,68,87,89,96,66,55,76,77,88,78,92,93);(1,3,6,14)(2,7,10,22)(4,12,17,28)(5,19,20,13)(8,21,15,29)(9,24,18,11)(16,31,32,27)(23,26,30,25)(33,35)(34,40)(36,45)(37,47)(38,48)(39,46)(41,55)(42,57)(43,58)(44,56)(49,66)(50,67)(51,68)(52,71)(53,69)(54,70)(59,65)(60,77)(61,78)(62,81)(63,79)(64,80)(72,76)(73,88)(74,89)(75,90)(82,86)(83,87)(84,93)(85,94)(91,92)(95,96);(1,4)(2,8)(3,12)(5,16)(6,17)(7,21)(9,23)(10,15)(11,25)(13,27)(14,28)(18,30)(19,31)(20,32)(22,29)(24,26)(33,36)(34,41)(35,45)(37,50)(38,51)(39,49)(40,55)(42,60)(43,61)(44,59)(46,66)(47,67)(48,68)(52,74)(53,72)(54,73)(56,65)(57,77)(58,78)(62,84)(63,82)(64,83)(69,76)(70,88)(71,89)(75,91)(79,86)(80,87)(81,93)(85,95)(90,92)(94,96)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,e^b=ei,d^c=di,f^c=fg74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36892);