Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35273 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,20,18,19,26,27)(15,21,22,24,23,28,25,16);(1,2)(3,4)(13,15,18,23)(14,16,19,24)(17,25,26,22)(20,28,27,21);(2,4)(6,9)(10,12)(13,16)(14,15)(17,21)(18,24)(19,23)(20,22)(25,27)(26,28);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,15)(17,21)(18,24)(19,23)(20,22)(25,27)(26,28);(14,19)(16,24)(20,27)(21,28)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6356188);