Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #263371 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
981 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)(3,5)(4,7)(6,8)(9,10,16,11)(12,20,22,24)(13,14,18,15)(17,23,19,21);(3,8)(5,6)(9,11,14,18,16,10,15,13)(12,17,21,20,22,19,23,24);(2,5)(3,8)(6,7)(9,12)(10,17)(11,19)(13,20)(14,21)(15,23)(16,22)(18,24);(9,13,14,11,16,18,15,10)(12,20,21,19,22,24,23,17)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fhi,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72305436, #72305442, #72305464, … |
| maximal normal subgroup of | #72305436, #72305442, #72305464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 170568);