Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394102 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1583 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,24)(21,23);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,21)(13,19)(14,20)(16,23)(17,22)(18,24);(10,16)(12,13)(15,22)(17,18)(19,21)(23,24);(1,4,5,3)(2,6,7,8)(10,17)(12,13)(15,22)(16,18)(19,21)(23,24)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72252475, #72252486, #72252500, … |
| maximal normal subgroup of | #72252475, #72252486, #72252500, … |
| maximal quotient of | C2×#395313 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302509);