Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #116168 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1439 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
699 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,10,23)(3,12)(8,22,26,30)(9,14,11,24)(13,27)(15,28)(17,32,25,21)(29,31);(2,8,9,25)(3,13)(4,16)(7,21,14,22)(10,26,11,17)(12,27)(15,28)(18,19)(23,32,24,30)(29,31);(1,2)(3,14)(4,17)(5,9)(6,11)(7,13)(8,19)(10,20)(12,23)(15,30)(16,25)(18,26)(21,28)(22,31)(24,27)(29,32)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=h,f^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,e^b=eh,f^b=fhi,d^c=dh,e^c=eh,f^c=fi,g^c=ghi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72233844, #72233845, #72233846, … |
| maximal normal subgroup of | #72233844, #72233845, #72233846, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23364);