Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219614 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23 | #540 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
449 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 449 conjugacy classes of subgroups — too many to draw. Only its 154 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 — 4 generators on 80 points:
(1,2,6,10,7,11,5,15)(3,8,13,16,12,9,4,14)(17,18)(19,30)(20,35,40,63)(21,41)(22,45)(23,28)(24,29)(25,33)(26,39,53,38)(27,42)(31,47,57,61)(32,58)(34,49)(36,64,66,51)(37,67,70,52)(43,54)(44,71)(46,50)(48,60)(55,59)(56,75,76,79)(62,69,78,68)(65,73,80,77)(72,74);(1,3,6,13,7,12,5,4)(2,8,10,16,11,9,15,14)(17,19,22,33,24,34,23,46)(18,25,28,30,29,50,45,49)(20,36,39,31,40,66,38,57)(21,32,42,59,44,60,43,72)(26,51,63,47,53,64,35,61)(27,48,41,55,54,58,71,74)(37,65,69,56,70,80,68,76)(52,75,78,73,67,79,62,77);(1,4,5,12,7,13,6,3)(2,9,10,14,11,8,15,16)(17,20,24,40)(18,26)(19,31)(21,37,44,70)(22,38,23,39)(25,47,50,61)(27,52)(28,35)(29,53)(30,51,49,64)(32,56)(33,36)(34,57)(41,62)(42,68,43,69)(45,63)(46,66)(48,73,58,77)(54,67)(55,75,74,79)(59,65)(60,76)(71,78)(72,80);(3,12)(4,13)(8,9)(14,16)(17,21,23,43,24,44,22,42)(18,27,45,71,29,54,28,41)(19,32,46,72,34,60,33,59)(20,37,39,69,40,70,38,68)(25,48,49,74,50,58,30,55)(26,52,63,78,53,67,35,62)(31,56,66,80,57,76,36,65)(47,73,64,79,61,77,51,75)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=efh,d^2=eh,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,f^b=fh,d^c=di,g^c=gh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 126876);