Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667484 |
| 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 | #23960 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2149 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)(13,14)(15,21)(16,20)(17,19)(18,23)(22,28)(24,26)(25,27);(6,10)(7,11)(8,12)(13,15,17,23)(14,18,19,21)(16,24,25,22)(20,28,27,26);(2,4)(5,6)(7,9)(8,11)(10,12);(14,19)(18,21)(20,27)(26,28);(5,7,9,6)(8,10,12,11)(13,16,17,25)(14,20,19,27)(15,22,23,24)(18,26,21,28)Polycyclic presentation:
2^9|b^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72283076, #72283364, #72283523, … |
| maximal normal subgroup of | #72283076, #72283364, #72283523, … |
| maximal quotient of | #72311312, #72311805, #72314373 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580513);