Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
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 | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1866 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)(3,5)(4,6)(7,8)(10,15,18,25)(12,22)(16,27,29,36)(17,30,31,23)(19,32)(24,37)(28,38,39,33)(34,40);(1,3,4,7)(2,5,6,8)(9,10)(11,16)(12,23)(13,25)(14,18)(15,26)(17,24)(19,33)(20,29)(21,36)(22,30)(27,35)(28,34)(31,37)(32,38)(39,40);(2,6)(5,8)(9,11)(10,16)(12,19)(13,21)(14,20)(15,27)(17,28)(18,29)(22,32)(23,33)(24,34)(25,36)(26,35)(30,38)(31,39)(37,40);(9,12,13,24,14,22,26,37)(10,17,15,23,18,31,25,30)(11,19,21,34,20,32,35,40)(16,28,27,33,29,39,36,38);(11,20)(16,29)(19,32)(21,35)(27,36)(28,39)(33,38)(34,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=di,d^b=df,f^b=fi,h^b=hi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264937, #72264955, #72265051, … |
| maximal normal subgroup of | #72264937, #72264955, #72265051, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287021);