Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35565 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2915 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,16,19,15,11,14,12)(13,17,24,21,22,18,23,20);(5,6)(7,8)(9,11,15,10)(12,16,19,14)(13,21,22,20)(17,23,18,24);(1,2,3,4)(6,8)(9,12,14,11,15,19,16,10)(13,20,23,18,22,21,24,17);(6,8)(9,13,14,23,15,22,16,24)(10,17,12,20,11,18,19,21);(13,22)(17,18)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72315498, #72315519, #72315550 |
| maximal normal subgroup of | #72315498, #72315519, #72315550 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351616);