Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535648 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36492 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6,10,9)(7,12,8,11)(13,14)(15,24)(16,27)(17,22)(18,21)(19,23)(20,25)(26,28);(5,7,10,8)(6,11,9,12)(13,15,18,26,19,16,17,25)(14,20,22,27,23,28,21,24);(2,4)(5,6,10,9)(7,12,8,11)(13,16,17,26,19,15,18,25)(14,20,21,24,23,28,22,27);(5,8,10,7)(6,11,9,12)(15,16)(20,28)(24,27)(25,26);(5,9,10,6)(7,12,8,11)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=f,d^2=hi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,d^c=di,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6461849);