Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| 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 | #60676 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1892 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)(3,6)(4,5)(7,8)(9,10,14,18,15,19,13,29)(11,20,24,30,21,35,23,16)(12,17,28,34,25,32,27,40)(22,36,38,33,26,39,37,31);(2,5)(6,8)(11,21)(12,25)(16,30)(17,32)(20,35)(23,24)(27,28)(34,40);(9,11,13,23,15,21,14,24)(10,16,18,20,19,30,29,35)(12,26,27,38,25,22,28,37)(17,33,34,39,32,31,40,36);(9,12,14,28,15,25,13,27)(10,17,18,34,19,32,29,40)(11,22,24,38,21,26,23,37)(16,31,20,36,30,33,35,39);(1,3,4,7)(2,6,5,8)(12,25)(17,32)(22,26)(27,28)(31,33)(34,40)(36,39)(37,38)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=h,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ci,d^b=di,d^c=di,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281172);