Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457579 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2840 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
969 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,8)(9,10)(11,17,19,14)(12,20)(13,16)(15,22)(18,24,21,23);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,21)(13,19)(15,24)(16,17)(18,22)(20,23);(1,3,4,7)(2,6,5,8)(9,12)(10,15,16,20)(11,18,19,21)(13,22)(14,23)(17,24);(3,7)(6,8)(9,11)(10,14)(12,21)(13,19)(15,23)(16,17)(18,22)(20,24)Polycyclic presentation:
2^9|a^2=fg,b^2=egh,c^2=ef,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,f^c=fi,g^c=gi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381381);