Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,11,9)(7,8,12,10)(21,22)(23,24)(25,27)(26,28);(5,7)(6,10)(8,9)(11,12)(14,17)(18,20)(21,23,25,28)(22,24,27,26);(1,2,3,4)(5,8,11,10)(6,12,9,7)(13,14,16,17)(15,18,19,20)(21,23)(22,26)(24,27)(25,28);(5,9,11,6)(7,8,12,10)(13,15,16,19)(14,18,17,20)(21,24,25,26)(22,23,27,28);(5,10,11,8)(6,12,9,7)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=h,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^c=df128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 424725);