Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437473 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(17,21)(18,24)(19,22)(20,23);(10,14)(11,18)(12,21)(16,17)(19,20)(22,24);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,20)(14,22)(16,23)(17,24)(18,21);(1,4,5,3)(2,6,7,8)(9,11)(10,15)(12,19)(13,20)(14,22)(16,23)(17,24)(18,21);(10,16)(14,17)(15,23)(22,24)Polycyclic presentation:
2^9|c^2=g,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72235088, #72235093, #72237063, … |
| maximal normal subgroup of | #72235088, #72235093, #72237063, … |
| maximal quotient of | #72308152, C2×#6451762 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6358958);