Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24464 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,18,15,19,13,27)(11,16,22,33,21,31,28,32)(12,23,26,35,25,17,29,36)(20,30,37,34,38,40,24,39);(1,3,5,4)(2,6,8,7)(9,11,14,22,15,21,13,28)(10,16,18,33,19,31,27,32)(12,24,26,20,25,37,29,38)(17,34,36,40,23,39,35,30);(1,3,5,4)(2,7,8,6)(9,12,13,29,15,25,14,26)(10,17,18,36,19,23,27,35)(11,20,28,24,21,38,22,37)(16,30,33,34,31,40,32,39);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=efg,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276571);