Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2861 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,7)(9,10,14,15)(11,16,12,19)(13,21,23,17)(18,22,20,24);(1,2,4,3)(6,7)(9,11)(10,16)(12,14)(13,22)(15,19)(17,18)(20,21)(23,24);(5,6,8,7)(9,12,14,11)(10,16,15,19)(13,20,23,18)(17,24,21,22);(5,7,8,6)(9,13,14,23)(10,17,15,21)(11,18,12,20)(16,24,19,22);(1,3,4,2)(9,10,14,15)(11,16,12,19)(13,17,23,21)(18,24,20,22)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470564);