Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397902 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| prime radical | #60639 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1883 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,18)(12,22)(13,16)(14,17)(15,20)(19,24)(21,23);(2,5)(6,7)(9,10)(11,18,19,24)(12,22)(13,16)(14,17)(15,21,23,20);(9,11,12,20,14,19,13,21)(10,15,16,18,17,23,22,24);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^9|b^2=eh,c^2=f,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72226719, #72249681, #72257232, … |
| maximal normal subgroup of | #72226719, #72249681, #72257232, … |
| maximal quotient of | #72257689, #72264067, #72302339 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323766);