Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #153520 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1408 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | #492 | C23 |
| prime radical | #19660 | C2 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C2×#492 ◁ G | |
| LECS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
309 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 309 conjugacy classes of subgroups — too many to draw. Only its 48 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)(3,12,15,25)(4,16,18,8)(5,10,21,9)(6,11)(7,14,22,30)(13,28,20,29)(17,24)(19,26,31,27)(23,32);(1,3)(2,7)(4,17)(5,20)(6,13)(8,25)(9,24)(10,23)(11,22)(12,26)(14,31)(15,21)(16,28)(18,30)(19,32)(27,29);(3,13)(7,22)(12,28)(14,30)(15,20)(17,32)(23,24)(25,29)Polycyclic presentation:
2^9|a^2=e,d^2=ghi,f^2=hi,g^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,c^b=ci,d^b=dgh,e^b=efgi,f^b=fhi,g^b=gi,e^d=egh,f^e=fhi,g^e=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72282994, #72283015, #72283064, … |
| maximal normal subgroup of | #72282994, #72283015, #72283064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60716);