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 | #52164 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14,20,16,19,23,18,28)(15,24,27,17,22,21,26,25);(5,7,10,6)(8,11,9,12)(13,15,18,26,19,22,20,27)(14,21,28,17,23,24,16,25);(2,4)(13,16,20,23,19,28,18,14)(15,25,27,24,22,17,26,21);(5,8,10,9)(6,11,7,12)(13,17,19,25)(14,22,23,15)(16,26,28,27)(18,24,20,21);(5,9,10,8)(6,11,7,12)(13,18,19,20)(14,16,23,28)(15,26,22,27)(17,21,25,24)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=h,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=df,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6303007);