Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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 | #36442 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2070 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)(7,11)(8,10)(9,12)(14,20)(15,22)(21,23)(25,26);(5,7,9,8)(6,10,12,11)(13,14,18,20)(15,24,22,28)(16,23,19,21)(17,26,27,25);(2,4)(13,15,18,22)(14,17,20,27)(16,25,19,26)(21,24,23,28);(5,7,9,8)(6,11,12,10)(13,16)(14,21)(15,22)(17,27)(18,19)(20,23);(5,8,9,7)(6,10,12,11)(13,17)(14,22)(15,20)(16,24)(18,27)(19,28)(21,26)(23,25)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6485725);