Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151066 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#367 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| prime radical | C23 | #367 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
131 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 131 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,22,28,24,30,20,26)(19,31,35,40,27,32,33,47)(21,38,39,48,41,36,37,45)(23,34,42,29,43,46,25,44);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,32,26,23,36,21,34,20,35,40,18,25,45,37,29,24,27,31,28,43,38,41,46,22,33,47,30,42,48,39,44);(3,9)(6,12)(10,15)(13,16)(17,20,24,22)(18,26,30,28)(19,33,27,35)(21,37,41,39)(23,25,43,42)(29,34,44,46)(31,47,32,40)(36,48,38,45)Polycyclic presentation:
2^9|a^2=e,b^2=dfh,c^2=i,d^2=fhi,e^2=g,f^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cg,e^b=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232029, #72234330, #72234333, … |
| maximal normal subgroup of | #72232029, #72234330, #72234333, … |
| maximal quotient of | #72321083 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58263);