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 | #35691 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2462 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)(13,14,18,21)(15,20,25,27)(16,22,17,19)(23,26,24,28);(2,4)(14,19)(15,23)(16,17)(21,22)(24,25)(26,28);(5,7,8,11)(6,10,9,12)(13,15)(14,20)(16,24)(17,23)(18,25)(19,26)(21,27)(22,28);(13,16)(14,21)(17,18)(20,26)(23,24)(27,28)Polycyclic presentation:
2^9|b^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72283209, #72283372, #72283982, … |
| maximal normal subgroup of | #72283209, #72283372, #72283982, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482758);