Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2749 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14519 |
| prime radical | #23399 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1303 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)(3,13,16,8)(4,17,18,31)(5,20)(6,12)(7,11)(9,19,27,32)(10,21)(14,24)(15,30,23,26)(22,28)(25,29);(1,3)(2,8)(4,14)(5,9)(6,23)(7,16)(10,18)(11,13)(12,30)(15,25)(17,24)(19,28)(20,32)(21,31)(22,27)(26,29);(1,4)(2,9,11,27)(3,14,16,10)(5,8)(6,19)(7,18)(12,17,29,31)(13,22)(15,20,23,28)(21,26)(24,30)(25,32);(1,5)(2,10)(3,9)(4,8)(6,24)(7,22)(11,14)(12,20)(13,18)(15,31)(16,27)(17,23)(19,30)(21,25)(26,32)(28,29)Polycyclic presentation:
2^9|a^2=h,c^2=eh,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286050, #72286176, #72286246, … |
| maximal normal subgroup of | #72286050, #72286176, #72286246, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295561);