Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| 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 | #16509 | C2 |
| 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)(5,6,9,12)(7,11,8,10)(13,14,16,18)(15,17,19,20)(22,24)(26,28);(7,8)(10,11)(13,15,16,19)(14,17,18,20)(21,22,25,24)(23,26,27,28);(5,7,9,8)(6,10,12,11)(13,15)(14,17)(16,19)(18,20)(21,23)(22,26)(24,28)(25,27);(1,2)(3,4)(5,8,9,7)(6,11,12,10);(5,6)(7,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,24,25,22)(23,26,27,28)Polycyclic presentation:
2^9|a^2=ghi,b^2=fgh,c^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4514515);