Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133338 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | C22×D4 | C22⋊C4 |
| LCS | G ▷ C22×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
504 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(17,18)(19,28,31,42)(20,25)(21,24)(22,27)(23,29,40,32)(26,36)(30,45,35,46)(33,43)(34,44)(37,41)(38,47,39,48);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,23)(20,32)(21,35)(22,31)(24,38)(25,42)(26,39)(27,40)(28,44)(29,34)(30,36)(33,47)(37,48)(41,45)(43,46);(18,24)(21,36)(23,38)(25,43)(26,27)(28,45)(30,35)(33,37)(39,40)(41,44)(42,46)(47,48)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,d^b=di,e^b=ehi,f^b=fgi,d^c=dh,e^c=ei,e^d=ei44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40796);