Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453981 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #15123 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1708 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(10,13)(14,16)(17,18,20,21)(19,22,24,23);(6,7)(9,10,12,13)(11,14,15,16)(17,19,20,24)(18,22,21,23);(5,7,8,6)(9,11,12,15)(10,14,13,16)(18,21)(19,23)(22,24);(2,4)(6,7)(11,15)(14,16)Polycyclic presentation:
2^9|a^2=i,b^2=ef,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=de,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72239750, #72239754, #72239759, … |
| maximal normal subgroup of | #72239750, #72239754, #72239759, … |
| maximal quotient of | #72257528 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363126);