Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219624 |
| 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 | #535 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
508 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 508 conjugacy classes of subgroups — too many to draw. Only its 162 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)(3,5)(4,11)(6,10)(7,14)(8,12)(9,15)(13,16)(17,18)(19,30)(20,36,40,51)(21,41)(22,28)(23,29)(24,33)(25,39,52,66)(26,43)(27,45)(31,58,61,46)(32,53)(34,48)(35,56)(37,49,60,76)(38,50,70,78)(42,47)(44,54)(55,62)(57,63)(59,77,73,75)(64,71)(65,80,72,69)(67,79,68,74);(1,3,5,2)(4,7,13,9)(6,10,12,8)(11,15,16,14)(17,19,22,33,23,35,45,34)(18,24,27,30,29,48,28,56)(20,37,66,31,40,60,39,61)(21,42,43,64,44,32,62,63)(25,49,36,46,52,76,51,58)(26,53,41,71,55,47,54,57)(38,68,80,73,70,67,69,59)(50,77,65,79,78,75,72,74);(1,4,12,15,5,13,6,14)(2,7,10,16,3,9,8,11)(17,20,23,40)(18,25)(19,31)(21,38,44,70)(22,39,45,66)(24,46,48,58)(26,50)(27,51)(28,36)(29,52)(30,49,56,76)(32,59)(33,37)(34,60)(35,61)(41,72)(42,73)(43,69,62,80)(47,74,53,79)(54,65)(55,78)(57,75,71,77)(63,67)(64,68);(2,3)(7,9)(8,10)(11,16)(17,21,22,43,23,44,45,62)(18,26,27,41,29,55,28,54)(19,32,33,63,35,42,34,64)(20,38,39,69,40,70,66,80)(24,47,30,57,48,53,56,71)(25,50,51,72,52,78,36,65)(31,59,37,67,61,73,60,68)(46,74,49,75,58,79,76,77)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=ef,d^2=ei,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=dhi,f^b=fh,g^c=gh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 126824);