Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3049 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)(14,18)(16,23)(19,24)(20,25)(22,27)(26,28);(5,6,10,9)(7,12,8,11)(13,14)(15,20)(16,24)(17,19)(18,23)(21,26)(22,28)(25,27);(2,4)(5,7,10,8)(6,11,9,12)(14,19)(15,21)(18,24)(22,27);(5,8,10,7)(6,11,9,12)(13,15,17,21)(14,20,19,26)(16,22,23,27)(18,25,24,28);(5,9,10,6)(7,12,8,11)(15,21)(20,26)(22,27)(25,28)Polycyclic presentation:
2^9|b^2=hi,c^2=hi,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,f^b=fi,d^c=di,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6479502);