Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14885 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3176 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)(5,6,10,11)(7,9,8,12)(13,14,18,16)(15,20,17,19)(23,27)(26,28);(6,11)(7,8)(13,15,18,17)(14,19,16,20)(22,25)(26,28);(5,7,10,8)(6,12,11,9)(13,14,18,16)(15,19,17,20)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,8,10,7)(6,12,11,9)(13,16,18,14)(15,19,17,20);(5,9,10,12)(6,7,11,8)(13,17,18,15)(14,19,16,20)(21,23)(22,26)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4286961);