Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2366 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
641 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,8,5,9,4,13)(3,7,11,16,12,10,14,15)(17,18,22,27,24,19,23,28)(20,29,30,26,31,32,21,25);(1,2,4,13,5,9,6,8)(3,10,14,16,12,7,11,15)(17,19)(18,24)(20,25)(21,29)(22,27)(23,28)(26,31)(30,32);(1,3)(2,7)(4,14)(5,12)(6,11)(8,16)(9,10)(13,15)(17,20,22,30,24,31,23,21)(18,25,28,32,19,26,27,29);(1,4,5,6)(2,8,9,13)(3,11,12,14)(7,15,10,16)(17,21,23,31,24,30,22,20)(18,26,28,29,19,25,27,32)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=fh,d^2=eh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ci,d^b=de,e^b=eh,g^b=gh,d^c=dhi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 138716);