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 | C22×D4 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1850 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)(6,10)(11,12)(13,14,17,21,18,20,26,27)(15,19,23,16,22,25,28,24);(5,6,9,10)(7,12,8,11)(15,22)(16,24)(19,25)(23,28);(2,4)(13,15)(14,19)(16,21)(17,23)(18,22)(20,25)(24,27)(26,28);(5,7,9,8)(6,11,10,12)(14,20)(15,22)(21,27)(23,28);(5,8,9,7)(6,11,10,12)(13,16,17,25,18,24,26,19)(14,15,21,23,20,22,27,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ci,d^b=df,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6292369);