Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395169 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6)(7,8)(10,15)(11,19)(12,13)(16,23)(17,18)(20,21);(1,2)(3,4)(9,10,13,17,14,18,12,15)(11,16,21,24,20,23,19,22);(2,4)(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,24)(18,23);(6,8)(9,10,12,15,14,18,13,17)(11,16,19,22,20,23,21,24)Polycyclic presentation:
2^9|b^2=ei,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^b=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72226912, #72226922, #72226937 |
| maximal normal subgroup of | #72226912, #72226922, #72226937 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302366);