Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32935 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,4)(5,6)(7,8)(9,10,14,18,15,19,13,30)(11,20,24,31,21,35,23,16)(12,25,29,34,27,40,28,17)(22,33,38,39,26,32,37,36);(9,11)(10,16,19,31)(12,26)(13,23)(14,24)(15,21)(17,33,34,32)(18,20,30,35)(22,27)(25,39,40,36)(28,38)(29,37);(1,2,3,4)(11,21)(12,27)(16,31)(17,34)(20,35)(23,24)(25,40)(28,29);(6,8)(9,12,13,28,15,27,14,29)(10,17,18,25,19,34,30,40)(11,22,23,37,21,26,24,38)(16,32,20,36,31,33,35,39)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=f,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,c^b=ci,d^b=di,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300396);