Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469944 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #32904 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(11,15)(14,16)(17,18,20,24,21,23,19,22);(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,22)(21,23);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(17,19,21,20)(18,22,23,24);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(18,23)(22,24);(9,11,12,15)(10,14,13,16)(17,19,21,20)(18,24,23,22)Polycyclic presentation:
2^9|a^2=f,c^2=gi,d^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dh,f^b=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6402940);