Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #145214 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2034 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #19187 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
300 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,10)(4,6)(5,9)(7,11)(8,13)(12,15)(14,16)(19,28)(20,33)(21,22)(24,38)(25,40)(26,37)(29,41)(30,45)(31,46)(32,34)(35,36)(39,48)(42,47)(43,44);(2,6)(3,11)(4,13)(7,15)(8,9)(10,14)(17,18)(19,29)(20,25)(21,37)(22,26)(23,27)(24,31)(28,44)(30,43)(32,39)(33,40)(34,38)(35,47)(36,42)(41,45)(46,48);(1,3,4,12,5,11,13,16)(2,7,8,15,9,10,6,14)(17,19,22,31,23,30,21,32)(18,24,26,43,27,39,37,29)(20,34,33,45,36,46,35,28)(25,41,40,48,42,44,47,38)Polycyclic presentation:
2^9|c^2=dh,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dg,f^b=fi,d^c=di,e^c=eghi,f^c=fgi,h^c=hi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52410);