Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #299233 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3455 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
846 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 846 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:
(1,2,6,9)(3,7,11,14)(4,13,8,5)(10,16,15,12)(17,18,23,28)(19,29,33,54)(20,25,37,49)(21,38,41,68)(22,42,39,27)(24,43,46,32)(26,50,51,40)(30,55,34,52)(31,59,61,45)(35,64,66,80)(36,67,56,48)(44,58,71,63)(47,72,73,65)(53,69,75,60)(57,70,62,78)(74,77,76,79);(1,3)(2,7)(4,10)(5,12)(6,11)(8,15)(9,14)(13,16)(17,19,22,32,23,33,39,43)(18,24,27,29,28,46,42,54)(20,34,36,63,37,30,56,58)(21,31,40,60,41,61,50,69)(25,44,48,52,49,71,67,55)(26,45,38,53,51,59,68,75)(35,62,65,79,66,57,72,77)(47,70,64,74,73,78,80,76);(1,4,5,2,6,8,13,9)(3,10,12,7,11,15,16,14)(17,20)(18,25)(19,30,33,34)(21,35)(22,36)(23,37)(24,44,46,71)(26,47)(27,48)(28,49)(29,52,54,55)(31,57,61,62)(32,58,43,63)(38,64)(39,56)(40,65)(41,66)(42,67)(45,70,59,78)(50,72)(51,73)(53,74,75,76)(60,77,69,79)(68,80);(3,11)(7,14)(10,15)(12,16)(17,21,39,50,23,41,22,40)(18,26,42,68,28,51,27,38)(19,31,43,69,33,61,32,60)(20,35,56,72,37,66,36,65)(24,45,54,75,46,59,29,53)(25,47,67,80,49,73,48,64)(30,57,63,79,34,62,58,77)(44,70,55,76,71,78,52,74)Polycyclic presentation:
2^9|a^2=hi,b^2=e,c^2=f,d^2=eh,e^2=h,f^2=i,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, 206434);