Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #263527 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23 | #535 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
725 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 725 conjugacy classes of subgroups — too many to draw. Only its 140 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)(3,7)(4,12)(5,14)(6,10)(8,13)(9,15)(11,16)(17,18,24,30)(19,31,35,57)(20,26,39,53)(21,40,44,55)(22,29,23,28)(25,45,49,34)(27,43,56,42)(32,50,63,59)(33,64,67,48)(36,47,62,76)(37,69,41,71)(38,58,46,52)(51,73,54,70)(60,74,80,66)(61,75,65,77)(68,79,72,78);(1,3,6,4)(2,7,10,12)(5,13,15,11)(8,9,16,14)(17,19,22,34,24,35,23,45)(18,25,28,31,30,49,29,57)(20,36,38,32,39,62,46,63)(21,33,42,66,44,67,43,74)(26,50,52,47,53,59,58,76)(27,48,40,60,56,64,55,80)(37,68,70,61,41,72,73,65)(51,78,69,75,54,79,71,77);(1,4,6,3)(2,8,10,16)(5,11,15,13)(7,14,12,9)(17,20)(18,26)(19,32)(21,41)(22,38)(23,46)(24,39)(25,47)(27,54)(28,52)(29,58)(30,53)(31,59)(33,65)(34,62)(35,63)(36,45)(37,44)(40,71)(42,73)(43,70)(48,77)(49,76)(50,57)(51,56)(55,69)(60,78)(61,67)(64,75)(66,68)(72,74)(79,80);(17,21,23,43,24,44,22,42)(18,27,29,55,30,56,28,40)(19,33,45,74,35,67,34,66)(20,37,46,73,39,41,38,70)(25,48,57,80,49,64,31,60)(26,51,58,71,53,54,52,69)(32,61,36,68,63,65,62,72)(47,75,50,78,76,77,59,79)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,g^b=gh,d^c=dh,f^c=fi,g^c=gh50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 170741);