Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2789 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6)(7,8)(10,15)(12,19)(13,21)(16,23)(17,22)(20,24);(1,2)(3,4)(5,6)(7,8)(9,10,13,15,14,17,21,22)(11,12,16,19,18,20,23,24);(6,8)(9,11,14,18)(10,12,17,20)(13,16,21,23)(15,19,22,24);(2,4)(6,8)(11,18)(12,20)(16,23)(19,24);(9,12,21,24,14,20,13,19)(10,16,22,11,17,23,15,18)Polycyclic presentation:
2^9|b^2=f,c^2=i,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72308113, #72308510, #72308890 |
| maximal normal subgroup of | #72308113, #72308510, #72308890 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426216);