Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343394 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3497 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3853 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,19,22,15)(12,23)(13,17)(14,18)(16,25)(20,28,36,33)(21,34,38,29)(24,31)(26,30)(27,32)(35,39,37,40);(2,5)(3,8)(6,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,28)(17,29)(18,19)(23,33)(24,35)(25,36)(26,37)(27,38)(30,39)(31,40)(32,34);(2,6)(5,7)(9,12,14,25)(10,16,18,23)(11,20,22,36)(13,26,27,24)(15,28,19,33)(17,31,32,30)(21,37,38,35)(29,40,34,39);(9,13,14,27)(10,17,18,32)(11,21,22,38)(12,24,25,26)(15,29,19,34)(16,30,23,31)(20,35,36,37)(28,39,33,40);(10,18)(15,19)(16,23)(17,32)(28,33)(29,34)(30,31)(39,40)Polycyclic presentation:
2^9|a^2=g,c^2=hi,d^2=hi,f^2=i,b^a=bf,c^a=ch,e^a=ehi,f^a=fh,f^b=fi,g^b=ghi,d^c=dhi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72281738, #72281794, #72281894 |
| maximal normal subgroup of | #72281738, #72281794, #72281894 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250864);