Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394022 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(10,15)(12,13)(16,18)(17,23)(19,21)(22,24);(1,3,4,8)(2,5,6,7)(10,16)(12,13)(15,18)(17,23)(19,21)(22,24);(9,10)(11,17)(12,16)(13,15)(14,18)(19,22)(20,24)(21,23);(9,11)(10,17)(12,21)(13,19)(14,20)(15,22)(16,23)(18,24)Polycyclic presentation:
2^9|a^2=e,b^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257746, #72257750, #72257815, … |
| maximal normal subgroup of | #72257746, #72257750, #72257815, … |
| maximal quotient of | #72264073, #72264078 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301635);