Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2646 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)(5,6,9,10)(7,8,12,11)(13,14,18,16)(15,20,17,19)(21,22)(23,27)(24,26)(25,28);(6,10)(7,12)(13,15,18,17)(14,19,16,20)(21,23,26,28)(22,27,24,25);(2,4)(14,16)(19,20)(22,24)(23,28);(1,2)(3,4)(5,7,9,12)(6,11,10,8)(13,16,18,14)(15,19,17,20)(21,24,26,22)(23,25,28,27);(5,8,9,11)(6,12,10,7)(13,17,18,15)(14,19,16,20)(21,25)(22,23)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4836865);