Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #363905 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13956 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62855 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2581 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,13,7,12,5,11)(3,9,15,23,16,25,20,8)(4,17,18,26,19,27,21,10)(14,28,31,24,30,22,29,32);(1,3)(2,8,12,23)(4,14)(5,15)(6,20)(7,16)(9,13,25,11)(10,22,26,28)(17,24,27,32)(18,29)(19,30)(21,31);(1,3)(2,9)(4,14)(5,20)(6,15)(7,16)(8,11)(10,24)(12,25)(13,23)(17,22)(18,31)(19,30)(21,29)(26,32)(27,28);(1,4)(2,10,12,26)(3,14)(5,18)(6,21)(7,19)(8,22,23,28)(9,24,25,32)(11,17,13,27)(15,29)(16,30)(20,31)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,h^b=hi,e^c=ei,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72233772, #72233778, #72233779, … |
| maximal normal subgroup of | #72233772, #72233778, #72233779, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271103);