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 | C2×#3487 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11050 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)(9,10,15,12)(11,18,13,16)(14,20,22,17)(19,23,21,24);(5,6)(7,8)(9,10,15,12)(11,16,13,18)(14,17,22,20)(19,23,21,24);(9,11,15,13)(10,16,12,18)(14,19,22,21)(17,23,20,24);(9,12,15,10)(11,16,13,18)(14,20,22,17)(19,23,21,24);(1,3,4,2)(9,13,15,11)(10,16,12,18)(14,21,22,19)(17,23,20,24);(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=h,e^2=i,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, 7532949);