Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #396 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(9,10,13,20,15,19,14,18)(11,21,23,16,24,33,25,32)(12,26,28,34,30,17,29,35)(22,37,38,31,27,36,39,40);(2,4)(9,11,14,25,15,24,13,23)(10,16,20,33,19,32,18,21)(12,27,29,38,30,22,28,39)(17,31,35,36,26,40,34,37);(6,8)(9,12,15,30)(10,17,19,26)(11,22,24,27)(13,29,14,28)(16,31,32,40)(18,35,20,34)(21,36,33,37)(23,39,25,38)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,f^b=fi,d^c=di,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72266464 |
| maximal normal subgroup of | #72266464 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30769);