Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| 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 | #32913 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1857 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)(5,6,8,9)(7,11,12,10)(14,19)(15,23)(16,25)(20,22);(5,6)(7,10)(8,9)(11,12)(13,14,17,19)(15,21,23,27)(16,26,25,28)(18,20,24,22);(2,4)(6,9)(10,11)(14,20)(15,23)(18,24)(19,22)(21,28)(26,27);(5,7,8,12)(6,10,9,11)(13,15,17,23)(14,21,19,27)(16,24,25,18)(20,26,22,28);(13,16,17,25)(14,21,19,27)(15,24,23,18)(20,28,22,26)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6578023);