Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 | |
4662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,13,16)(11,12,17,15)(14,18,23,24)(19,22,20,21);(2,4)(9,11,13,17)(10,15,16,12)(14,19,23,20)(18,21,24,22);(6,8)(10,16)(11,19)(12,21)(14,23)(15,22)(17,20);(9,12,13,15)(10,17,16,11)(14,22,23,21)(18,20,24,19);(11,19)(12,22)(14,23)(15,21)(17,20)(18,24)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72313380, #72313623, #72313664, … |
| maximal normal subgroup of | #72313380, #72313623, #72313664, … |
| maximal quotient of | #72315713 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480657);