Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | #2397 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
400 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9)(4,12,14,7)(5,11)(6,8)(10,13,16,15)(17,18,21,26,23,25,29,41,22,27,36,42,20,34,35,38)(19,28,32,44,33,39,45,48,30,43,46,47,31,24,37,40);(1,3,9,2,6,11,5,8)(4,10,13,7,14,16,15,12)(17,19,29,45,20,31,21,32,22,30,35,37,23,33,36,46)(18,24,38,47,34,43,42,48,27,39,41,44,25,28,26,40);(1,4)(2,7)(3,10)(5,15)(6,14)(8,12)(9,13)(11,16)(19,30)(24,39)(28,43)(31,33)(32,46)(37,45)(40,48)(44,47)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,c^b=ci,e^b=egi,d^c=dgi,f^c=fgi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72239338, #72308622 |
| maximal normal subgroup of | #72239338, #72308622 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45102);