Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| 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 | #60749 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1946 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)(9,10,15,21,14,19,30,20)(11,22,17,12,24,16,31,26)(13,18,29,37,28,35,38,36)(23,34,33,25,27,32,40,39);(1,3)(2,5)(4,8)(6,7)(9,11,15,17,14,24,30,31)(10,16,21,26,19,22,20,12)(13,27,29,40,28,23,38,33)(18,34,37,25,35,32,36,39);(9,12,15,16,14,26,30,22)(10,17,21,24,19,31,20,11)(13,25,29,32,28,39,38,34)(18,33,37,27,35,40,36,23);(3,8)(5,7)(9,13)(10,18)(11,23)(12,25)(14,28)(15,29)(16,32)(17,33)(19,35)(20,36)(21,37)(22,34)(24,27)(26,39)(30,38)(31,40);(2,6)(5,7)(10,19)(11,24)(13,28)(17,31)(20,21)(25,39)(29,38)(32,34)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,g^c=gi,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6301876);