Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395154 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2011 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(9,10)(11,14)(12,13)(15,16)(18,19)(20,24)(21,22);(5,6)(7,8)(10,13)(14,16)(17,18,21,19,23,20,22,24);(1,2,3,4)(6,8)(17,19,21,20,23,24,22,18);(9,11,12,15)(10,14,13,16)(17,20,21,24,23,18,22,19)Polycyclic presentation:
2^9|b^2=h,c^2=fi,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280207, #72280820, #72280913 |
| maximal normal subgroup of | #72280207, #72280820, #72280913 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313597);