Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393899 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(14,22)(17,23);(1,3,4,7)(2,5,6,8)(9,10)(11,17)(12,20)(13,15)(14,19)(16,21)(18,23)(22,24);(9,11,12,19,13,18,21,24)(10,14,16,17,15,22,20,23);(2,6)(5,8)Polycyclic presentation:
2^9|b^2=h,c^2=g,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,g^b=gi,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72224747, #72226738, #72226741, … |
| maximal normal subgroup of | #72224747, #72226738, #72226741, … |
| maximal quotient of | #72237225, #72264144, #72264397 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 317162);