Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61569 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
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×C4⋊C4 |
| prime radical | C22×#41 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,4)(3,10,12,16)(5,9,6,15)(8,11,14,13)(17,18,21,25,22,26,20,24)(19,27,28,32,30,31,29,23);(1,3,6,13,7,12,5,11)(2,8,9,10,4,14,15,16)(17,19,20,29,22,30,21,28)(18,23,25,27,26,32,24,31);(1,4,7,2)(3,8,12,14)(5,9,6,15)(10,11,16,13)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,27,32,31);(1,5,7,6)(2,9,4,15)(3,11,12,13)(8,10,14,16)(17,20,22,21)(18,24,26,25)(19,29,30,28)(23,31,32,27);(3,12)(8,14)(10,16)(11,13)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=h,d^2=h,f^2=h,g^2=i,b^a=bf,f^a=fh,c^b=ch,e^b=ehi,d^c=dhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251647);