Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9369 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | #532 | C2×C4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
312 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 86 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:
(1,2,5,9,6,8,4,14)(3,10,13,15,12,16,11,7)(17,18,22,28,23,29,43,56,24,30,44,57,21,27,41,54)(19,25,33,47,32,46,63,74,34,48,65,76,39,52,69,78)(20,35,38,67,37,50,68,55,36,53,42,70,40,26,49,51)(31,58,61,79,62,45,71,73,59,66,64,80,60,72,77,75);(1,3)(2,7,8,15)(4,11)(5,13)(6,12)(9,10,14,16)(17,19,22,33,23,32,43,63,24,34,44,65,21,39,41,69)(18,25,28,47,29,46,56,74,30,48,57,76,27,52,54,78)(20,31,42,64,37,62,49,77,36,59,38,61,40,60,68,71)(26,45,55,75,35,66,70,79,50,72,51,73,53,58,67,80);(2,8)(4,5)(7,15)(11,13)(17,20)(18,26)(19,31)(21,40)(22,42)(23,37)(24,36)(25,45)(27,53)(28,55)(29,35)(30,50)(32,62)(33,64)(34,59)(38,44)(39,60)(41,68)(43,49)(46,66)(47,75)(48,72)(51,57)(52,58)(54,67)(56,70)(61,65)(63,77)(69,71)(73,76)(74,79)(78,80)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=g,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,e^b=egi,d^c=dgi,e^c=ei,f^c=fgi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45083);