Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #147512 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10960 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
958 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 958 conjugacy classes of subgroups — too many to draw. Only its 156 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:
(1,2,5,10)(3,12,14,23)(4,16,18,8)(6,11,20,9)(7,15,22,31)(13,27,29,32)(17,24,19,25)(21,28,26,30);(1,3)(2,7,11,23)(4,13)(5,14,20,31)(6,15)(8,21,24,27)(9,22)(10,12)(16,26)(17,29,18,28)(19,30)(25,32);(1,4)(2,8)(3,13)(5,17,20,18)(6,19)(7,21)(9,25,10,16)(11,24)(12,26,22,32)(14,29,31,28)(15,30)(23,27)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gh,h^a=hi,d^b=dhi,e^b=ehi,f^b=fhi,h^b=hi,d^c=dhi,e^c=eh,f^c=fh,h^c=hi,f^d=fi,g^d=gi,f^e=fi,g^e=gi,g^f=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72238712, #72238719, #72238721, … |
| maximal normal subgroup of | #72238712, #72238719, #72238721, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54708);