Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2076 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)(13,14)(15,26)(16,28)(17,22)(18,23)(19,24)(20,25)(21,27);(5,6)(7,10)(8,9)(11,12)(13,15,18,27,20,16,19,22)(14,21,24,26,25,17,23,28);(2,4)(6,9)(10,12)(13,16,18,22,20,15,19,27)(14,21,24,26,25,17,23,28);(5,7)(6,10)(8,11)(9,12)(13,17,20,21)(14,22,25,27)(15,23,16,24)(18,28,19,26);(7,11)(10,12)(13,18,20,19)(14,23,25,24)(15,27,16,22)(17,26,21,28)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,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, 6447061);