Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #348090 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3531 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
747 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 747 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 256 |
A minimal faithful permutation representation — 4 generators on 80 points:
(2,6)(4,11)(7,15)(8,12)(9,13)(14,16)(17,18,24,30)(19,31,36,48)(20,26,40,53)(21,41,43,56)(22,29,23,28)(25,35,49,34)(27,45,54,42)(32,57,37,66)(33,61,63,47)(38,68,70,72)(39,55,44,52)(46,64,50,60)(51,69,76,71)(58,65,74,62)(59,73,67,75)(77,78,80,79);(17,19,23,35,24,36,22,34)(18,25,29,48,30,49,28,31)(20,37,44,60,40,32,39,64)(21,33,45,65,43,63,42,62)(26,50,55,57,53,46,52,66)(27,47,56,74,54,61,41,58)(38,67,71,78,70,59,69,79)(51,75,68,77,76,73,72,80);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15)(17,20)(18,26)(19,32)(21,38)(22,39)(23,44)(24,40)(25,46)(27,51)(28,52)(29,55)(30,53)(31,57)(33,59)(34,60)(35,64)(36,37)(41,72)(42,69)(43,70)(45,71)(47,73)(48,66)(49,50)(54,76)(56,68)(58,77)(61,75)(62,78)(63,67)(65,79)(74,80);(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)(17,21)(18,27)(19,33)(20,38)(22,42)(23,45)(24,43)(25,47)(26,51)(28,41)(29,56)(30,54)(31,58)(32,59)(34,62)(35,65)(36,63)(37,67)(39,69)(40,70)(44,71)(46,73)(48,74)(49,61)(50,75)(52,72)(53,76)(55,68)(57,77)(60,78)(64,79)(66,80)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,f^c=fi,g^c=ghi,f^d=fhi,g^d=gh53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 255291);