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×#459 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1916 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,10,11,12)(14,18)(19,21)(20,22)(27,28);(6,9)(10,12)(13,14)(15,22)(16,18)(17,19)(20,25)(21,26)(23,27)(24,28);(1,2,3,4)(14,19)(15,23)(17,26)(18,21)(20,22)(24,25);(5,7,8,11)(6,10,9,12)(13,15,16,25)(14,20,18,22)(17,24,26,23)(19,27,21,28);(14,19)(15,24)(17,26)(18,21)(23,25)(27,28)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^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 | #72263918, #72263937, #72263979 |
| maximal normal subgroup of | #72263918, #72263937, #72263979 |
| maximal quotient of | #72264211, #72265166 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6490814);