Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,18)(15,17,19,20)(22,25)(26,27);(13,15)(14,17)(16,19)(18,20)(21,22,24,25)(23,27,28,26);(6,9)(10,12)(15,19)(17,20)(21,23,24,28)(22,26,25,27);(2,4)(23,28)(26,27);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^c=df104 conjugacy classes — view the full character table.
| maximal subgroup of | #72303657, #72303658, #72303680, … |
| maximal normal subgroup of | #72303657, #72303658, #72303680, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 518221);