Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2362 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)(5,6,10,9)(7,12,8,11)(14,17)(15,19)(22,26)(24,28);(5,7,10,8)(6,11,9,12)(13,14,16,17)(15,18,19,20)(22,26)(23,27);(5,7,10,8)(6,12,9,11)(13,15,16,19)(14,18,17,20)(21,22)(23,24)(25,26)(27,28);(5,8,10,7)(6,12,9,11)(13,14)(15,18)(16,17)(19,20)(21,23)(22,24)(25,27)(26,28);(5,9,10,6)(7,12,8,11)(13,15)(14,18)(16,19)(17,20)(21,24)(22,23)(25,28)(26,27)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=ef104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 484356);