Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62283 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60453 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,20,15,21,30,19)(11,16,12,24,23,17,26,32)(13,18,29,36,28,34,39,35)(22,31,25,38,37,33,27,40);(1,3)(2,5)(4,8)(6,7)(9,11,14,12,15,23,30,26)(10,16,20,24,21,17,19,32)(13,22,29,25,28,37,39,27)(18,31,36,38,34,33,35,40);(9,12,14,23,15,26,30,11)(10,17,19,24,21,16,20,32)(13,27,29,22,28,25,39,37)(18,31,35,40,34,33,36,38);(3,8)(5,7)(9,13)(10,18)(11,22)(12,25)(14,29)(15,28)(16,31)(17,33)(19,35)(20,36)(21,34)(23,37)(24,38)(26,27)(30,39)(32,40);(2,6)(5,7)(13,28)(18,34)(22,37)(25,27)(29,39)(31,33)(35,36)(38,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,d^c=di,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72308222, #72308757, #72308766 |
| maximal normal subgroup of | #72308222, #72308757, #72308766 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6296397);