Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13312 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #469 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9078 |
| prime radical | #3067 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
421 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 421 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(1,2,4,8,6,12,21,24,7,13,22,25,5,11,20,10)(3,14,15,30,16,29,32,23,18,31,19,28,17,9,26,27);(2,8,12,24,13,25,11,10)(9,27,14,30,29,23,31,28);(1,3)(2,9)(4,19)(5,17)(6,16)(7,18)(8,23)(10,30)(11,31)(12,14)(13,29)(15,22)(20,32)(21,26)(24,28)(25,27)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=cgh,d^c=di,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214259, #72214352, #72214383, … |
| maximal normal subgroup of | #72214259, #72214352, #72214383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44860);