Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1762 |
| prime radical | C23 | #418 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
325 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 325 conjugacy classes of subgroups — too many to draw. Only its 108 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:
(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18,21,28)(19,32)(20,29,38,46)(22,26,39,27)(23,31,24,30)(25,40)(33,43)(34,44)(35,47)(36,41)(37,45)(42,48);(1,3)(2,6,8,15)(4,11)(5,10)(7,9,12,16)(13,14)(17,19,22,33,24,36,38,40,21,37,39,48,23,35,20,34)(18,25,29,41,31,43,27,32,28,44,46,47,30,42,26,45);(3,10)(6,15)(9,16)(11,14)(17,20,24,22,21,38,23,39)(18,26,31,29,28,27,30,46)(19,33,35,34,37,48,36,40)(25,41,42,45,44,47,43,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=eg,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fg,g^a=gi,h^a=hi,c^b=cg,e^b=ei,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72287493, #72287513, #72287538, … |
| maximal normal subgroup of | #72287493, #72287513, #72287538, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35802);