Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131842 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8443 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#3291 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
799 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 799 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:
(1,2,5,12,22,8,23,11,7,13,4,19,6,10,20,29)(3,14,17,28,15,31,16,9,18,27,21,25,24,30,32,26);(2,8)(4,20)(5,23)(6,22)(9,25)(10,13)(12,19)(14,27)(15,24)(16,17)(21,32)(26,28);(1,3)(2,9)(4,21)(5,17)(6,24)(7,18)(8,25)(10,28)(11,30)(12,27)(13,26)(14,19)(15,22)(16,23)(20,32)(29,31)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=egi,f^b=fghi,g^b=gi,d^c=dgi,e^c=eg,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72286402 |
| maximal normal subgroup of | #72286402 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39039);