Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454426 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
869 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,5)(3,8,4,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,24,22,20,30,25)(19,23,28,32,29,27,31,26);(1,3,6,4)(2,7,5,8)(9,11,12,15)(10,13,14,16)(17,19,21,28,22,29,30,31)(18,23,24,32,20,27,25,26);(1,4,6,3)(2,7,5,8)(10,14)(13,16)(17,18)(19,26)(20,22)(21,25)(23,31)(24,30)(27,28)(29,32);(1,5,6,2)(3,8,4,7)(17,20)(18,22)(19,27)(21,25)(23,29)(24,30)(26,28)(31,32)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,c^2=e,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368271);