Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133341 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | C23 | #379 |
| LCS | G ▷ C22×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
574 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 574 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) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(17,18)(19,27)(20,30)(21,25)(22,32)(23,28)(24,29)(26,31);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,23,22)(18,24,28,30)(20,31,29,21)(25,27,26,32);(17,20,23,29)(18,25,28,26)(19,21,22,31)(24,32,30,27)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=i,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,c^b=ci,d^b=di,e^b=ehi,f^b=fgi,d^c=dh,e^c=ei,e^d=ei50 conjugacy classes — view the full character table.
| maximal subgroup of | #72267666 |
| maximal quotient of | #72267644 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40804);