Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,10,9)(7,8,11,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,24)(25,27)(26,28);(6,9)(7,11)(14,18)(16,19)(21,23,25,28)(22,24,27,26);(5,7)(6,8)(9,12)(10,11)(13,15,17,20)(14,19,18,16)(21,23)(22,26)(24,27)(25,28);(1,2)(3,4)(5,8,10,12)(6,11,9,7)(21,24,25,26)(22,23,27,28);(5,9,10,6)(7,12,11,8)(13,16,17,19)(14,20,18,15)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=fgh,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 559182);