Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1786 |
| prime radical | #3061 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
599 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,5,3,8,6)(9,10,14,18,15,12,25,19)(11,20,22,34,24,37,23,39)(13,28,30,17,31,35,40,27)(16,32,33,38,26,29,36,21);(1,3)(2,5)(4,6)(7,8)(9,11)(10,16)(12,26)(13,29)(14,22)(15,24)(17,34)(18,33)(19,36)(20,28)(21,30)(23,25)(27,39)(31,32)(35,37)(38,40);(2,3)(6,7)(9,12)(10,15)(11,16)(13,27)(14,18)(17,31)(19,25)(20,21)(22,36)(23,33)(24,26)(28,40)(29,34)(30,35)(32,39)(37,38)Polycyclic presentation:
2^9|a^2=f,d^2=gi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=dg,g^b=gi,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36759);