Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
853 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(10,15,18,29)(11,19)(12,25,27,28)(16,20,33,30)(22,37,35,39)(23,24)(32,38)(36,40);(1,3,4,8)(2,5,6,7)(9,10,14,18)(11,20,19,30)(12,26,27,31)(13,29,21,15)(16,23,33,24)(17,28,34,25)(22,38,35,32)(36,37,40,39);(2,6)(5,7)(9,11,21,24,14,19,13,23)(10,16,15,30,18,33,29,20)(12,22,25,39,27,35,28,37)(17,32,31,40,34,38,26,36);(3,8)(5,7)(9,12)(10,17)(11,22)(13,28)(14,27)(15,31)(16,32)(18,34)(19,35)(20,36)(21,25)(23,37)(24,39)(26,29)(30,40)(33,38)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=efh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296618);