Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #244523 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3487 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1060 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 1060 conjugacy classes of subgroups — too many to draw. Only its 144 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:
(1,2,5,8)(3,6,10,12)(4,7,11,14)(9,15,16,13)(17,18)(19,30)(20,36)(21,42)(22,46)(23,28)(24,29)(25,33)(26,40)(27,43)(31,59)(32,48)(34,50)(35,49)(37,57)(38,68)(39,54)(41,53)(44,56)(45,55)(47,61)(51,65)(52,71)(58,62)(60,77)(63,74)(64,73)(66,69)(67,72)(70,76)(75,80)(78,79);(6,12)(13,15)(17,19,24,35)(18,25,29,50)(20,37,41,65)(21,32,45,63)(22,34,23,33)(26,51,54,57)(27,48,56,74)(28,49,46,30)(31,40,61,39)(36,47,53,59)(38,67,71,79)(42,58,55,73)(43,64,44,62)(52,75,68,77)(60,69,80,70)(66,72,76,78);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,16)(14,15)(17,20,24,41)(18,26,29,54)(19,31,35,61)(21,38,45,71)(22,39,23,40)(25,47,50,59)(27,52,56,68)(28,53,46,36)(30,57,49,51)(32,60,63,80)(33,37,34,65)(42,66,55,76)(43,69,44,70)(48,72,74,78)(58,77,73,75)(62,79,64,67);(1,4)(2,7)(3,9)(5,11)(6,13)(8,14)(10,16)(12,15)(17,21)(18,27)(19,32)(20,38)(22,44)(23,43)(24,45)(25,48)(26,52)(28,55)(29,56)(30,58)(31,60)(33,64)(34,62)(35,63)(36,66)(37,67)(39,70)(40,69)(41,71)(42,46)(47,72)(49,73)(50,74)(51,75)(53,76)(54,68)(57,77)(59,78)(61,80)(65,79)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,f^c=fh,g^c=gh,e^d=eh,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151730);