Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #237595 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #60798 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1695 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,20)(13,16)(14,18)(15,22)(19,24)(21,23);(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17);(2,6)(3,4)(7,8)(12,22)(14,23)(15,20)(18,21);(1,3,5,4)(2,7,8,6)Polycyclic presentation:
2^9|b^2=h,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,e^b=eh,d^c=di,e^c=eh,f^c=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72338676, #72338828, #72338831 |
| maximal normal subgroup of | #72338676, #72338828, #72338831 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 144804);