Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#447 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1799 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,13)(15,16)(17,18,20,21)(19,22,24,23);(1,3)(2,5)(4,8)(6,7)(17,19)(18,22)(20,24)(21,23);(2,6)(5,7)(10,13)(14,16)(18,21)(19,23)(22,24);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,18,20,21)(19,22,24,23);(3,8)(5,7)(18,21)(19,23)(22,24)Polycyclic presentation:
2^9|a^2=fi,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249676, #72305296, #72305302, … |
| maximal normal subgroup of | #72249676, #72305296, #72305302, … |
| maximal quotient of | #72249788, #72305711, C2×#6699796 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6606992);