Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30434 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14391 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,11,5,20)(3,12,15,8,14,24,21,22)(4,16,19,25,18,30,17,9)(13,23,28,29,27,32,31,26);(8,22)(12,24)(13,27)(28,31);(1,3)(2,8)(4,13)(5,21)(6,15)(7,14)(9,23)(10,24)(11,22)(12,20)(16,29)(17,31)(18,27)(19,28)(25,32)(26,30);(1,4,5,17,7,18,6,19)(2,9,10,16,11,25,20,30)(3,13,21,31,14,27,15,28)(8,23,24,29,22,32,12,26)Polycyclic presentation:
2^9|a^2=h,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72237064, #72237067, #72237094, … |
| maximal normal subgroup of | #72237064, #72237067, #72237094, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292029);