Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| automorphism | Aut(G) order 1048576 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2170 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)(5,6,10,8)(7,12,9,11)(13,14,19,18)(15,16,20,17)(21,22,25,27)(23,28,26,24);(5,7,10,9)(6,11,8,12)(13,15,19,20)(14,17,18,16)(23,26)(24,28);(7,9)(11,12)(13,16,19,17)(14,15,18,20)(21,23,25,26)(22,24,27,28);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(13,17,19,16)(14,20,18,15);(5,9,10,7)(6,12,8,11)(13,18,19,14)(15,17,20,16)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=f,e^2=f,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, 4029915);