Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
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 | C23 | C8×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
316 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 100 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:
(1,2,5,8,6,9,13,15,4,3,7,10,12,11,14,16)(17,18)(19,28)(20,32,35,42)(21,36)(22,26)(23,27)(24,30)(25,34,41,33)(29,44,46,38)(31,39)(37,40)(43,45,48,47);(1,3,5,10,6,11,13,16,4,2,7,8,12,9,14,15)(17,19,21,30,23,31,22,37)(18,24,26,28,27,40,36,39)(20,29,34,47,35,46,33,45)(25,38,42,43,41,44,32,48);(2,3)(8,10)(9,11)(15,16)(17,20,23,35)(18,25,27,41)(19,29,31,46)(21,34,22,33)(24,38,40,44)(26,42,36,32)(28,43,39,48)(30,47,37,45)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=g,d^2=g,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ci,e^b=eg,d^c=dg,e^c=eg,f^c=fg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45027);