Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437608 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2621 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,16)(12,23)(13,19)(14,18)(15,20)(17,24)(21,22);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,19)(13,23)(14,22)(15,17)(18,21)(20,24);(9,12)(10,17)(11,20)(13,24)(14,16)(15,21)(18,23)(19,22);(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,22)(23,24);(9,13,15,14)(10,18,20,19)(11,22,17,23)(12,24,21,16)Polycyclic presentation:
2^9|a^2=f,b^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72227838, #72227843, #72227844 |
| maximal normal subgroup of | #72227838, #72227843, #72227844 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345360);