Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61569 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C2×#3482 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,10,6,8,4,9)(3,11,13,7,14,16,12,15)(17,18,20,24,21,23,27,30)(19,22,26,29,25,28,31,32);(1,3,6,14)(2,7,8,15)(4,12,5,13)(9,11,10,16);(2,8)(7,15)(9,10)(11,16)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32);(1,4,6,5)(2,9,8,10)(3,12,14,13)(7,11,15,16)(19,25)(22,28)(26,31)(29,32);(18,23)(22,28)(24,30)(29,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,e^a=ehi,f^a=fh,f^b=fh,d^c=dhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250338);