Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573511 |
| 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 | |
4890 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)(9,10,14,15)(11,13,17,22)(12,16,19,23)(18,20,21,24);(2,4)(10,15)(11,13)(12,19)(17,22)(18,24)(20,21);(6,8)(9,11)(10,13)(12,20)(14,17)(15,22)(16,21)(18,23)(19,24);(9,12,14,19)(10,16,15,23)(11,18,17,21)(13,20,22,24);(9,13,14,22)(10,17,15,11)(12,21,19,18)(16,24,23,20)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^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 | #72308549, #72308890 |
| maximal normal subgroup of | #72308549, #72308890 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480717);