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 | #3455 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
549 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 549 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) = 80 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 80 points:
(1,2)(3,10,12,7)(4,6)(5,9)(8,13)(11,15,16,14)(17,18,21,26,22,27,36,46,23,28,37,47,38,48,59,69)(19,24,31,41,32,42,55,65,30,40,53,63,54,64,74,78)(20,25,34,44,35,45,58,68,33,43,56,66,57,67,75,79)(29,49,51,70,52,71,73,80,50,39,60,61,72,62,76,77);(2,6)(4,13)(7,14)(8,9)(10,15)(11,16)(17,19)(18,24)(20,29)(21,31)(22,32)(23,30)(25,39)(26,41)(27,42)(28,40)(33,50)(34,51)(35,52)(36,55)(37,53)(38,54)(43,49)(44,61)(45,62)(46,65)(47,63)(48,64)(56,60)(57,72)(58,73)(59,74)(66,70)(67,71)(68,77)(69,78)(75,76)(79,80);(1,3,5,12)(2,7,9,10)(4,11,13,16)(6,14,8,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,33)(24,39)(26,44)(27,45)(28,43)(30,50)(31,51)(32,52)(36,58)(37,56)(38,57)(40,49)(41,61)(42,62)(46,68)(47,66)(48,67)(53,60)(54,72)(55,73)(59,75)(63,70)(64,71)(65,77)(69,79)(74,76)(78,80)Polycyclic presentation:
2^9|a^2=f,c^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, 36905);