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 | C2×#2456 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2878 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)(7,9)(8,11)(10,12)(13,14,16,17)(15,19,20,18)(22,26)(24,27);(6,7)(10,11)(14,17)(18,19)(21,22)(23,24)(25,26)(27,28);(5,6,9,7)(8,11,12,10)(13,15,16,20)(14,18,17,19)(21,23,25,28)(22,27,26,24);(5,7,9,6)(8,11,12,10)(13,14)(15,18)(16,17)(19,20)(21,24)(22,23)(25,27)(26,28);(5,8,9,12)(6,10,7,11)(13,15)(14,18)(16,20)(17,19)(21,23)(22,24)(25,28)(26,27)Polycyclic presentation:
2^9|a^2=i,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, 484355);