Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152684 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13083 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | #23998 | C2 |
| LCS | G ▷ C22×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #117 ◁ #3508 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
484 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 484 conjugacy classes of subgroups — too many to draw. Only its 64 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)(3,7,13,21)(4,15)(5,10,19,9)(6,11)(8,16)(12,26,27,32)(14,22,18,25)(17,23,29,24)(20,30,31,28);(1,3,6,14)(2,7,10,21,11,22,9,25)(4,12,16,28)(5,18,19,13)(8,20,24,31,15,26,23,32)(17,30,29,27);(1,4,6,16)(2,8)(3,12,14,28)(5,17,19,29)(7,20)(9,23)(10,24)(11,15)(13,27,18,30)(21,31)(22,26)(25,32)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dgi,f^b=fgi,g^b=gh,e^c=ei,f^d=fhi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72339873, #72339880 |
| maximal normal subgroup of | #72339873, #72339880 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59880);