Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5377 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,7)(9,10,13,12)(11,15,16,14)(17,18,20,23)(19,22,24,21);(10,12)(14,15)(17,19)(18,21)(20,24)(22,23);(5,6,8,7)(11,16)(14,15)(19,24)(21,22);(1,2,3,4)(5,7,8,6)(9,11,13,16)(10,14,12,15)(17,19)(18,22)(20,24)(21,23);(9,12,13,10)(11,15,16,14)(17,18)(19,22)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232230, #72232243, #72232253, … |
| maximal normal subgroup of | #72232230, #72232243, #72232253, … |
| maximal quotient of | #72279040, #72279228, #72299053, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4250227);