Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397902 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
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 | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1595 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)(3,7)(4,5)(6,8)(14,21)(17,23);(2,5)(6,7)(9,10)(11,17,18,23)(12,16)(13,15)(14,19,21,24)(20,22);(9,11,12,19,13,18,20,24)(10,14,22,23,15,21,16,17);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^9|b^2=eh,c^2=fg,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72252539, #72252542, #72252546, … |
| maximal normal subgroup of | #72252539, #72252542, #72252546, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324330);