Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,11)(9,12)(13,14,16,17)(15,18,20,19)(21,22,26,25)(23,24,28,27);(5,6,10,11)(7,9,8,12)(14,17)(18,19)(21,23)(22,27)(24,25)(26,28);(5,7,10,8)(6,12,11,9)(13,15,16,20)(14,18,17,19)(21,23,26,28)(22,27,25,24);(1,2)(3,4)(5,8,10,7)(6,9,11,12)(13,14)(15,18)(16,17)(19,20)(21,24)(22,23)(25,28)(26,27);(5,9)(6,8)(7,11)(10,12)(13,15)(14,19)(16,20)(17,18)(21,25,26,22)(23,27,28,24)Polycyclic presentation:
2^9|a^2=fgh,b^2=gh,c^2=f,d^2=gh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4032552);