Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8528 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1779 |
| prime radical | #3531 | C4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
519 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 519 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 256 |
A minimal faithful permutation representation — 3 generators on 80 points:
(2,6)(4,11)(7,15)(8,12)(9,13)(14,16)(17,18,22,29,24,28,45,55,23,27,44,54,21,39,42,67)(19,30,33,56,32,47,63,73,34,25,43,48,40,49,68,74)(20,35,38,64,37,26,50,51,36,53,65,78,41,52,69,77)(31,58,61,79,62,46,70,72,59,71,66,80,60,57,75,76);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15)(17,19)(18,25)(20,31)(21,40)(22,43)(23,34)(24,32)(26,46)(27,30)(28,49)(29,56)(33,44)(35,58)(36,59)(37,62)(38,66)(39,47)(41,60)(42,63)(45,68)(48,54)(50,75)(51,76)(52,57)(53,71)(55,73)(61,65)(64,80)(67,74)(69,70)(72,77)(78,79);(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)(17,20,23,36)(18,26,27,52)(19,31,34,59)(21,41,24,37)(22,38,44,65)(25,46,30,57)(28,53,39,35)(29,51,54,77)(32,62,40,60)(33,61,43,66)(42,69,45,50)(47,58,49,71)(48,72,56,76)(55,78,67,64)(63,70,68,75)(73,79,74,80)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dgi,e^b=eg,f^b=fi,d^c=dg,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36867);