Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #432893 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15468 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1026 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)(3,12,14,7)(4,16)(5,9)(6,10)(8,18)(11,21)(13,22,17,25)(15,24,20,23)(19,27)(26,30)(28,31,29,32);(1,3)(2,7)(4,17,18,13)(5,20)(6,14)(8,25,16,22)(9,24)(10,12)(11,23)(15,21)(19,28,30,29)(26,31,27,32);(1,4)(2,8,10,16)(3,13)(5,19)(6,18)(7,22,12,25)(9,26,11,27)(14,17)(15,28)(20,29)(21,30)(23,31,24,32);(2,9)(5,21)(7,23)(8,26)(10,11)(12,24)(15,20)(16,27)(19,30)(22,31)(25,32)(28,29)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,f^c=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72319143, #72319342, #72319444, … |
| maximal normal subgroup of | #72319143, #72319342, #72319444, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340089);