Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453981 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,3,4)(9,10,12,13)(11,14,15,16)(17,18,23,26,22,28,21,27)(19,24,31,32,20,30,29,25);(5,6)(7,8)(10,13)(14,16)(17,19,22,20)(18,24,28,30)(21,29,23,31)(25,26,32,27);(2,4)(6,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,19)(18,25,28,32)(21,29,23,31)(24,26,30,27);(11,15)(14,16)(18,26)(21,23)(24,32)(25,30)(27,28)(29,31)Polycyclic presentation:
2^9|a^2=efi,b^2=i,c^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361845);