Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #513092 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | #2215 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
298 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 114 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:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,29)(20,35)(21,24)(22,41)(23,44)(25,32)(26,36)(27,43)(28,38)(30,58)(31,55)(33,65)(34,57)(37,70)(39,56)(40,68)(42,54)(45,53)(46,61)(47,63)(48,66)(49,64)(50,75)(51,59)(52,79)(60,77)(62,80)(67,72)(69,74)(71,76)(73,78);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(18,24)(25,46)(26,35)(27,47)(28,48)(29,55)(41,56)(44,68)(49,51)(50,76)(52,67)(53,77)(54,78)(57,58)(65,80)(70,74);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,21,31)(18,25,24,46)(20,30,36,34)(22,42,39,73)(23,45,40,60)(26,49,35,51)(27,53,47,77)(28,54,48,78)(29,56,55,41)(32,63,61,43)(33,66,62,38)(37,71,69,75)(44,50,68,76)(52,80,67,65)(57,70,58,74)(59,79,64,72);(2,8)(6,9)(7,16)(14,15)(17,20,21,36)(18,26,24,35)(19,30,31,34)(22,37,39,69)(23,38,40,66)(25,49,46,51)(27,52,47,67)(28,44,48,68)(29,57,55,58)(32,59,61,64)(33,60,62,45)(41,74,56,70)(42,71,73,75)(43,72,63,79)(50,78,76,54)(53,80,77,65)Polycyclic presentation:
2^9|b^2=ei,c^2=e,d^2=i,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,h^a=hi,c^b=cei,f^c=fg,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420336);