Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573511 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2744 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)(5,6)(7,8)(10,15)(11,18)(16,21)(17,20);(1,2,3,4)(9,10,13,15)(11,19,18,22)(12,21,23,16)(14,20,24,17);(6,8)(9,11,13,18)(10,12,15,23)(14,21,24,16)(17,22,20,19);(9,12,13,23)(10,16,15,21)(11,20,18,17)(14,22,24,19);(10,17)(12,22)(14,24)(15,20)(16,21)(19,23)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72284913, #72285147, #72307072, … |
| maximal normal subgroup of | #72284913, #72285147, #72307072, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481100);