Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573477 |
| 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 |
| prime radical | #36152 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5213 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)(3,4)(5,6)(7,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,23)(21,24);(2,4)(11,19)(12,20)(13,21)(15,22)(17,23)(18,24);(6,8)(9,11,14,12)(10,15,16,17)(13,20,21,19)(18,23,24,22);(10,16)(11,19)(12,20)(13,21)(15,23)(17,22);(9,12,14,11)(10,17,16,15)(13,19,21,20)(18,22,24,23)Polycyclic presentation:
2^9|c^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,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 | #72282753, #72283488, #72283577, … |
| maximal normal subgroup of | #72282753, #72283488, #72283577, … |
| maximal quotient of | #72307772, #72307832, #72308025, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480678);