Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19755 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | D4×C4⋊C4 |
| prime radical | C2×#135 | Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,3,10,8)(4,11,14,12,15,7,13,16)(17,18,23,27)(19,28,25,21)(20,31,26,22)(24,30,32,29);(1,3)(2,6)(4,12)(5,8)(7,14)(9,10)(11,13)(15,16)(17,19)(18,24)(20,28)(21,26)(22,30)(23,25)(27,32)(29,31);(17,20,22,18,23,26,31,27)(19,21,30,32,25,28,29,24);(1,4,6,15)(2,7,3,11)(5,14,10,13)(8,12,9,16)(17,21,23,28)(18,25,27,19)(20,30,26,29)(22,32,31,24)Polycyclic presentation:
2^9|a^2=g,c^2=f,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,c^b=ci,d^b=de,e^b=eh,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 133403);