Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×#120 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,7,8)(10,15,18,17)(11,19,21,22);(1,2)(3,4)(9,10,13,15,14,18,12,17)(11,20,19,23,21,16,22,24);(2,4)(9,11,12,19,14,21,13,22)(10,16,15,23,18,20,17,24);(9,12,14,13)(10,17,18,15)(11,19,21,22)(16,24,20,23)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,c^b=cg,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72210005 |
| maximal normal subgroup of | #72210005 |
| maximal quotient of | #72210001, #72239944, #72240021, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265458);