Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #316187 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2504 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #124 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
449 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 449 conjugacy classes of subgroups — too many to draw. Only its 122 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)(17,18,24,28,23,30,22,29)(19,31,35,48,36,55,34,25)(20,37,39,52,40,51,44,26)(21,41,45,54,43,53,42,27)(32,49,59,56,61,46,60,73)(33,62,64,75,65,47,63,57)(38,50,67,66,68,74,70,69)(58,77,78,72,80,76,79,71);(1,3,5,2)(4,9,13,7)(6,10,12,8)(11,14,16,15)(17,19,22,34,23,36,24,35)(18,25,28,31,30,48,29,55)(20,32,39,59,40,61,44,60)(21,33,42,63,43,65,45,64)(26,49,51,73,52,46,37,56)(27,47,41,57,54,62,53,75)(38,58,67,78,68,80,70,79)(50,72,69,77,74,71,66,76);(1,4,12,15,5,13,6,14)(2,7,10,16,3,9,8,11)(17,20,23,40)(18,26)(19,32)(21,38,43,68)(22,39,24,44)(25,46,48,49)(27,50)(28,51)(29,37)(30,52)(31,56,55,73)(33,58)(34,59)(35,60)(36,61)(41,69)(42,67,45,70)(47,71,62,72)(53,66)(54,74)(57,76,75,77)(63,78)(64,79)(65,80);(2,3)(7,9)(8,10)(11,16)(17,21,24,45,23,43,22,42)(18,27,29,53,30,54,28,41)(19,33,35,64,36,65,34,63)(20,38,44,70,40,68,39,67)(25,47,55,75,48,62,31,57)(26,50,37,66,52,74,51,69)(32,58,60,79,61,80,59,78)(46,71,73,77,49,72,56,76)Polycyclic presentation:
2^9|a^2=egi,b^2=e,c^2=ef,d^2=ehi,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,f^b=fh,g^c=gh50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 223395);