Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| socle | C22 | C2×#342 |
| prime radical | C24 | #95 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
615 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,8,6,7,5,3)(9,10,13,18,14,16,28,32)(11,19,22,35,20,36,26,39)(12,23,24,34,25,17,27,33)(15,29,30,37,31,38,40,21);(1,3,5,7,6,8,4,2)(9,11)(10,15,16,31)(12,21)(13,26)(14,20)(17,19,23,36)(18,30,32,40)(22,28)(24,38)(25,37)(27,29)(33,35,34,39);(2,7)(3,8)(10,16)(11,20)(17,23)(18,32)(21,37)(22,26)(29,38)(33,34)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=i,e^2=h,f^2=hi,b^a=bd,c^a=ch,d^a=dg,f^a=fi,g^a=gi,c^b=ch,e^b=egi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34343);