Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393934 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#3485 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
4336 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)(3,4)(9,10,14,17)(11,15,20,23)(12,13,16,21)(18,24,22,19);(5,6)(7,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24);(2,4)(6,8)(9,12,14,16)(10,13,17,21)(11,18,20,22)(15,19,23,24);(9,13)(10,16)(11,19)(12,17)(14,21)(15,22)(18,23)(20,24)Polycyclic presentation:
2^9|a^2=i,c^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,h^b=hi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72238043, #72239358, #72256600, … |
| maximal normal subgroup of | #72238043, #72239358, #72256600, … |
| maximal quotient of | #72238044, #72238045, #72249937, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301265);