Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149743 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11499 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | #381 | C23 |
| prime radical | C22×H(5,𝔽2) | C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ #2308 ◁ G | |
| LECS | G ▷ #381 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
1107 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 1107 conjugacy classes of subgroups — too many to draw. Only its 60 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)(4,15,18,27)(6,10,19,28)(7,20,21,16,22,31,26,12)(8,23,25,17);(1,3)(2,7)(4,16)(5,13)(6,14)(8,24)(9,26)(10,22)(11,15)(12,17)(18,31)(19,30)(20,23)(21,28)(25,29)(27,32);(4,17)(8,15)(11,24)(12,16)(18,23)(20,31)(25,27)(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,d^b=di,e^b=efi,g^b=gi,d^c=di,e^d=eh,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72237387, #72237729, #72237730, … |
| maximal normal subgroup of | #72237387, #72237729, #72237730, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56939);