Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2139 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,10)(11,12)(13,14,18,15,17,19,26,21)(16,20,25,22,24,27,28,23);(5,6)(7,11)(8,12)(9,10)(13,15,18,19,17,21,26,14)(16,23,25,20,24,22,28,27);(1,2,3,4)(5,7,9,8)(6,11,10,12)(14,19)(15,21)(20,27)(22,23);(5,8,9,7)(6,11,10,12)(13,16,17,24)(14,20,19,27)(15,22,21,23)(18,25,26,28);(16,24)(20,27)(22,23)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ci,d^b=df,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6290126);