Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #59130 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2798 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(6,7)(9,10,14,18)(11,19,21,24)(12,17,13,16)(15,22,23,20);(1,2)(3,4)(6,7)(9,11,12,22,14,21,13,20)(10,15,16,24,18,23,17,19);(5,6,8,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,19,23,24);(2,4)(5,7,8,6)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,19,23,24);(9,11,12,22,14,21,13,20)(10,15,16,24,18,23,17,19)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470464);