Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3264 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2325 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12);(6,9)(10,12)(13,14)(15,22)(16,18)(17,21)(19,25)(20,26)(23,27)(24,28);(2,4)(5,7,8,11)(6,10,9,12)(14,18)(15,23)(16,26)(19,22)(20,21)(24,25);(13,15)(14,19)(16,24)(17,25)(18,27)(20,28)(21,22)(23,26);(14,18)(15,24)(16,26)(20,21)(23,25)(27,28)Polycyclic presentation:
2^9|c^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,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 | #72250199, #72256956, #72256979, … |
| maximal normal subgroup of | #72250199, #72256956, #72256979, … |
| maximal quotient of | #72264513, #72264757, #72265286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481988);