Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62842 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C22×#537 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11086 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,3)(6,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23);(5,6,7,8)(11,17)(15,21)(18,23)(22,24);(1,3,4,2)(10,14)(15,21)(16,19)(22,24);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24);(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=i,e^2=h,b^a=bg,c^a=ch,f^a=fh,e^b=eh,d^c=dh164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532912);