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 | |
4156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,18)(12,23)(13,16)(14,17)(15,21)(19,22)(20,24);(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,24)(18,23);(1,3,5,4)(2,6,7,8)(9,10)(11,19)(12,16)(13,23)(14,17)(15,21)(18,22)(20,24);(1,4,5,3)(2,6,7,8)(9,10)(11,19)(12,16)(13,23)(14,17)(15,21)(18,22)(20,24);(9,12,14,13)(10,16,17,23)(11,20,22,21)(15,19,24,18)Polycyclic presentation:
2^9|c^2=g,d^2=g,e^2=i,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 | #72233137, #72260459, #72261029, … |
| maximal normal subgroup of | #72233137, #72260459, #72261029, … |
| maximal quotient of | #72233129, #72233848, #72260660 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6358961);