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 | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2137 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)(5,6,8,10)(7,11,12,9)(15,22)(19,27)(23,24)(26,28);(5,7,8,12)(6,9,10,11)(13,14,17,21)(15,19,22,27)(16,18,25,20)(23,28,24,26);(2,4)(14,18)(15,23)(16,25)(19,27)(20,21)(22,24);(13,15)(14,19)(16,24)(17,22)(18,26)(20,28)(21,27)(23,25);(14,18)(15,24)(16,25)(20,21)(22,23)(26,28)Polycyclic presentation:
2^9|a^2=f,b^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=di,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 | #72250261, #72251258, #72256879, … |
| maximal normal subgroup of | #72250261, #72251258, #72256879, … |
| maximal quotient of | #72264176, #72264697, #72308334, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481619);