Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #383150 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14357 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60367 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1969 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11)(3,12,16,29)(4,17)(5,21,24,10)(7,22,26,15)(8,20)(9,23)(13,28)(14,25)(18,30)(19,27)(31,32);(1,3)(2,7)(4,18)(5,22)(6,16)(8,25)(9,28)(10,29)(11,26)(12,21)(13,20)(14,19)(15,24)(17,32)(23,31)(27,30);(1,4)(2,8)(3,13)(5,19)(6,20)(7,25)(9,21)(10,27)(11,17)(12,28)(14,15)(16,18)(22,31)(23,24)(26,32)(29,30);(1,4)(2,9)(3,14)(5,23)(6,20)(7,25)(8,21)(10,17)(11,27)(12,30)(13,15)(16,31)(18,22)(19,24)(26,32)(28,29)Polycyclic presentation:
2^9|a^2=h,e^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,f^b=fi,h^c=hi,e^d=ei,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72318354, #72318900, #72318969, … |
| maximal normal subgroup of | #72318354, #72318900, #72318969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290346);