Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11502 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator, prime radical | C24 | C22×C8 |
| Frattini, cosocle | C2×C22⋊C8 | C23 |
| socle | C22 | C2×#323 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
401 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 401 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(1,2,3,7,4,8,9,14,5,6,10,12,11,13,15,16)(17,18)(19,27)(20,24)(21,34,35,25)(22,26)(23,29,37,44)(28,41)(30,42)(31,45)(32,47,48,39)(33,40)(36,43,38,46);(17,19,22,30)(18,23,26,37)(20,31,33,28)(21,29,35,44)(24,38,40,36)(25,27,34,42)(32,46,48,43)(39,45,47,41);(2,6)(7,12)(8,13)(14,16)(17,20,22,33)(18,24,26,40)(19,28,30,31)(21,32,35,48)(23,36,37,38)(25,39,34,47)(27,41,42,45)(29,43,44,46)Polycyclic presentation:
2^9|a^2=e,b^2=h,c^2=h,e^2=g,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,c^b=ch,e^b=ef,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29429);