Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993979 |
| 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 | #24372 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18,20,23,21,24,19,22);(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,14)(13,16)(17,19,21,20)(18,22,24,23);(9,11)(10,13)(12,16)(14,15)(17,18)(19,23)(20,22)(21,24);(1,3,4,8)(2,6,5,7)(11,16)(13,15)(17,19,21,20)(18,22,24,23);(10,14)(13,15)(17,19,21,20)(18,23,24,22)Polycyclic presentation:
2^9|a^2=g,b^2=fi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6914526);