Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
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 | #61303 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#446 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,7)(9,10,14,18,15,19,13,28)(11,20,23,16,21,32,29,30)(12,17,27,35,24,33,26,40)(22,36,37,31,25,38,39,34);(1,2,4,5)(3,7,8,6)(11,21)(12,24)(16,30)(17,33)(20,32)(23,29)(26,27)(35,40);(9,11)(10,16)(12,25)(13,29)(14,23)(15,21)(17,34)(18,32)(19,30)(20,28)(22,24)(26,37)(27,39)(31,33)(35,36)(38,40);(1,3,4,8)(2,6,5,7);(9,12,14,27,15,24,13,26)(10,17,18,35,19,33,28,40)(11,22,23,37,21,25,29,39)(16,31,32,38,30,34,20,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=f,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ci,d^b=df,e^b=ei,e^c=ei,g^c=gi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291804);