Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417496 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #35347 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2821 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)(5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,19)(20,23)(22,24);(9,11,13,19)(10,14,16,18)(12,17,20,24)(15,22,23,21);(2,4)(9,12)(10,15)(11,17)(13,20)(14,21)(16,23)(18,22)(19,24);(6,8)(9,10)(11,18)(12,15)(13,16)(14,19)(17,22)(20,23)(21,24)Polycyclic presentation:
2^9|b^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=dei,e^c=ei,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72234970, #72261428 |
| maximal normal subgroup of | #72234970, #72261428 |
| maximal quotient of | #72234115 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324725);