Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,11)(7,8,12,10)(14,18)(16,20)(21,22,26,23)(24,28,25,27);(5,7,9,12)(6,10,11,8)(13,14,17,18)(15,16,19,20)(21,23,26,22)(24,27,25,28);(5,7)(6,8)(9,12)(10,11)(13,15,17,19)(14,16,18,20)(21,24,26,25)(22,27,23,28);(1,2)(3,4)(21,25,26,24)(22,28,23,27);(5,8)(6,7)(9,10)(11,12)(13,16)(14,19)(15,18)(17,20)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=fh,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 754312);