Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| 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 | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1071 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,11,24,10)(3,13,17,28,18,27,16,8)(4,19,21,23,9,29,20,5)(14,22,26,15,30,25,32,31);(1,3)(2,8)(4,14)(5,22)(6,17)(7,18)(9,30)(10,27)(11,28)(12,13)(15,19)(16,24)(20,32)(21,26)(23,25)(29,31);(1,4)(2,5)(3,14)(6,21)(7,9)(8,25)(10,29)(11,23)(12,19)(13,31)(15,27)(16,32)(17,26)(18,30)(20,24)(22,28);(1,5,7,23)(2,9,11,4)(3,15)(6,19,24,29)(8,26)(10,21,12,20)(13,30)(14,27)(16,22)(17,25)(18,31)(28,32)Polycyclic presentation:
2^9|a^2=h,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319859, #72319908, #72320000, … |
| maximal normal subgroup of | #72319859, #72319908, #72320000, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292058);