Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190034 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator | C24 | C23×C4 |
| Frattini, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1205 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(9,10)(11,14,18,22)(12,19,20,15)(13,16)(17,23)(21,24);(1,2,4,6)(3,7,8,5)(9,11)(10,14,16,22)(12,17)(13,18)(15,21,19,23)(20,24);(1,3,4,8)(2,5,6,7)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23);(9,11)(10,14,16,22)(12,17)(13,18)(15,21,19,23)(20,24)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=g,d^2=f,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^b=hi,h^c=hi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270261);