Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C25×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(10,12)(14,15)(17,18)(19,23)(20,22)(21,24);(5,6,8,7)(9,10,13,12)(11,15,16,14)(17,18,20,22)(19,21,24,23);(1,2,4,3)(5,7,8,6)(17,18)(19,21)(20,22)(23,24);(1,3,4,2)(9,11,13,16)(10,14,12,15)(17,18)(19,21)(20,22)(23,24);(9,12,13,10)(11,15,16,14)(17,19,20,24)(18,21,22,23)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df116 conjugacy classes — view the full character table.
| maximal subgroup of | #72240718, #72240774, #72240778, … |
| maximal normal subgroup of | #72240718, #72240774, #72240778, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 438419);