Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13578 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#353 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | #382 | C23 |
| prime radical | C22×#127 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2285 ◁ G | |
| LECS | G ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
646 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 646 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,5,4,6,7,8)(9,10,12,17)(11,15,19,22)(13,16,20,23)(14,18,21,24);(10,15)(11,13)(12,20)(16,18)(19,21)(22,23);(9,11,14,13)(10,15,18,16)(12,19,21,20)(17,22,24,23)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=h,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=dh,e^b=efhi,f^b=fi,e^d=ehi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72293179, #72338320, #72338389, … |
| maximal normal subgroup of | #72293179, #72338320, #72338389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56765);