Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514000 |
| 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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(9,10)(11,15)(12,21)(13,22)(14,19)(16,18)(17,24)(20,23);(1,2,4,3)(5,6)(7,8)(9,11,15,10)(12,22,13,21)(14,18,16,19)(17,24,20,23);(9,12,15,13)(10,17,11,20)(14,23,16,24)(18,22,19,21);(6,8)(9,13,15,12)(10,17,11,20)(14,24,16,23)(18,22,19,21);(1,3,4,2)(9,14,15,16)(10,18,11,19)(12,23,13,24)(17,22,20,21)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426333);