Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7440 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1353 |
| commutator | C2×M16 | C42 |
| Frattini, cosocle | C2×#423 | C22 |
| socle | C2 | #7251 |
| prime radical | C24 | #104 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#423 ◁ G | |
| LECS | G ▷ C2×#423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
273 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 273 conjugacy classes of subgroups — too many to draw. Only its 60 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 — 2 generators on 32 points:
(1,2)(3,11,13,24)(4,15)(5,18)(6,10)(7,12,21,29)(8,17)(9,20)(14,27,19,28)(16,26)(22,30,23,32)(25,31);(1,3)(2,7,9,22,10,21,18,23)(4,12,16,29)(5,19)(6,13)(8,24,15,28,25,11,26,27)(14,20)(17,32,31,30)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,e^2=hi,f^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,h^a=hi,c^b=cg,d^b=di,f^b=fi,g^b=gh,e^c=eh,g^c=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72339762, #72339769, #72339770 |
| maximal normal subgroup of | #72339762, #72339769, #72339770 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1532);