Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393893 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,4)(10,14)(11,12)(15,16);(2,4)(5,6)(7,8)(9,10)(11,16)(12,14)(13,15);(6,8)(9,10)(11,14)(12,16)(13,15);(11,12)(14,16)Polycyclic presentation:
2^9|e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^b=hi,f^c=fi,h^c=hi,e^d=ei,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257992, #72264462, #72264501, … |
| maximal normal subgroup of | #72257992, #72264462, #72264501, … |
| maximal quotient of | #72258074, #72283262 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301089);