Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7348 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1317 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | #407 | C23 |
| socle | C2 | C2×#1317 |
| prime radical | C25 | M16 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#407 ◁ G | |
| LECS | G ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
436 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 436 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(1,2,5,11,6,10,19,26,7,12,18,28,4,9,17,25)(3,13,15,29,14,8,21,23,16,22,30,32,20,24,31,27);(1,3)(2,8,12,24)(4,14)(5,15)(6,20)(7,16)(9,22,10,13)(11,27,28,23)(17,21)(18,30)(19,31)(25,29,26,32);(1,4,7,6)(2,9,12,10)(3,14,16,20)(5,17,18,19)(8,22,24,13)(11,25,28,26)(15,21,30,31)(23,32,27,29)Polycyclic presentation:
2^9|a^2=e,b^2=h,c^2=i,d^2=i,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,e^b=efi,g^b=gi,e^d=eh,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72338322, #72338357 |
| maximal normal subgroup of | #72338322, #72338357 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56750);