Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32898 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
4044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10)(11,15)(12,20,22,16)(13,17)(14,18)(19,23,24,21);(1,2,3,4)(9,11,14,17)(10,15,18,13)(12,19,22,24)(16,23,20,21);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20);(6,8)(9,13,14,15)(10,17,18,11)(12,21,22,23)(16,24,20,19)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,d^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,f^b=fi,e^c=ei,h^c=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72251665, #72251698, #72251719, … |
| maximal normal subgroup of | #72251665, #72251698, #72251719, … |
| maximal quotient of | #72262634 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299857);