Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149763 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11502 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | #381 | C23 |
| prime radical | C2×#525 | C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ C2×#381 ◁ G | |
| LECS | G ▷ #381 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
471 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 471 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9)(3,11,13,29,14,24,30,32)(4,15,17,27)(6,10,19,28)(7,20,22,16,21,31,26,12)(8,23,25,18);(1,3)(2,7)(4,16)(5,13)(6,14)(8,24)(9,26)(10,21)(11,15)(12,18)(17,31)(19,30)(20,23)(22,28)(25,29)(27,32);(7,21)(11,24)(22,26)(29,32)Polycyclic presentation:
2^9|a^2=e,d^2=i,e^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ch,d^b=di,e^b=efi,g^b=gi,e^d=eh,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72237020, #72237387, #72237388, … |
| maximal normal subgroup of | #72237020, #72237387, #72237388, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56959);