Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433044 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35256 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2019 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,25)(19,23,27,30)(20,29,24,21)(26,32,31,28);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,26)(21,28)(22,27)(24,31)(25,30)(29,32);(11,15)(13,16)(17,20,21,18,22,24,29,25)(19,26,28,23,27,31,32,30);(1,3,4,7)(2,6,5,8)(10,14)(13,16);(19,27)(23,30)(26,31)(28,32)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,e^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341060);