Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12310 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1619 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8443 |
| prime radical | #61105 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
794 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 794 conjugacy classes of subgroups — too many to draw. Only its 112 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:
(1,2,6,12,10,13,24,11,8,14,5,22,7,4,20,21)(3,15,17,27,26,9,25,28,19,16,32,30,18,29,23,31);(1,3)(2,9)(4,16)(5,23)(6,25)(7,26)(8,19)(10,18)(11,28)(12,30)(13,15)(14,29)(17,24)(20,32)(21,31)(22,27);(1,4)(2,10)(3,16)(5,11)(6,21)(7,14)(8,13)(9,18)(12,24)(15,19)(17,30)(20,22)(23,28)(25,31)(26,29)(27,32)Polycyclic presentation:
2^9|a^2=f,d^2=h,e^2=gi,f^2=ghi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,h^a=hi,d^b=dh,e^b=eg,f^b=fghi,g^b=gi,d^c=dg,e^c=eg,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225390, #72225399 |
| maximal normal subgroup of | #72225390, #72225399 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39030);