Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6,9,10)(7,12,11,8)(13,14,17,19)(15,20,16,18)(22,25)(23,27);(1,3,4,2)(6,10)(8,12)(13,15,17,16)(14,18,19,20);(6,10)(7,11)(15,16)(18,20)(21,22,24,25)(23,26,27,28);(5,7,9,11)(6,8,10,12)(13,16,17,15)(14,18,19,20)(21,23)(22,26)(24,27)(25,28);(5,8)(6,11)(7,10)(9,12)(13,14)(15,20)(16,18)(17,19)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,c^2=fgh,d^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 443311);