Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514000 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36260 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2909 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,22)(20,24)(21,23);(1,2,4,3)(6,7)(9,11)(10,14)(12,19)(13,18)(15,17)(16,24)(20,21)(22,23);(5,6,8,7)(10,15)(14,17)(16,23)(22,24);(5,7,8,6)(10,15)(11,18)(13,21)(14,22)(17,24)(19,20);(1,3,4,2)(10,15)(11,19)(16,23)(18,20)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,e^b=ei,h^b=hi,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72283437, #72284945, #72315498 |
| maximal normal subgroup of | #72283437, #72284945, #72315498 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470304);