Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393958 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35223 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6)(7,8)(9,10)(11,18)(12,16)(13,15)(14,19)(17,21)(20,24)(22,23);(1,2,4,3)(9,11,12,19)(10,14,16,18)(13,22,21,20)(15,24,17,23);(1,3,4,2)(10,15)(13,21)(14,23)(16,17)(18,24)(20,22);(6,8)Polycyclic presentation:
2^9|b^2=e,c^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72261564, #72261573, #72261580, … |
| maximal normal subgroup of | #72261564, #72261573, #72261580, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301903);