Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346446 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62820 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2394 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)(3,13,16,25)(4,17,21,9)(5,10,7,12)(8,18,24,14)(15,29,22,32)(19,27,20,26)(23,30,28,31);(1,3)(2,8)(4,18)(5,22)(6,16)(7,15)(9,25)(10,28)(11,24)(12,23)(13,17)(14,21)(19,30)(20,31)(26,32)(27,29);(4,19)(9,26)(13,29)(14,31)(17,27)(18,30)(20,21)(25,32);(1,2,7,12)(3,8,15,23)(4,17,19,27)(5,10,6,11)(9,20,26,21)(13,30,29,18)(14,25,31,32)(16,24,22,28)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,e^d=eh59 conjugacy classes — view the full character table.
| maximal subgroup of | #72213838, #72232843, #72232980, … |
| maximal normal subgroup of | #72213838, #72232843, #72232980, … |
| maximal quotient of | #72284391, #72285896, #72338704, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253658);