Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #391960 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14666 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,8,13,23)(4,10)(5,9)(6,12)(7,11)(14,22,29,32)(15,24,16,21)(17,26,30,31)(18,27)(19,28)(20,25);(1,3,6,16,7,13,5,15)(2,8,9,24,11,23,12,21)(4,17,18,14,19,30,20,29)(10,26,25,32,28,31,27,22);(2,9)(3,13)(5,6)(8,21)(10,27)(11,12)(17,30)(18,20)(22,31)(23,24)(25,28)(26,32);(1,4)(2,10)(3,14)(5,20)(6,18)(7,19)(8,22)(9,25)(11,28)(12,27)(13,29)(15,17)(16,30)(21,31)(23,32)(24,26)Polycyclic presentation:
2^9|a^2=g,b^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,f^c=fi,h^c=hi,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237047, #72237059, #72237068, … |
| maximal normal subgroup of | #72237047, #72237059, #72237068, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299158);