Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| 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 | |
2456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,11,12,9)(14,16)(15,20)(22,26)(23,28);(1,2,3,4)(13,14,17,16)(15,19,20,18)(21,22)(23,24)(25,26)(27,28);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,18)(16,19)(17,20)(21,23)(22,27)(24,26)(25,28);(5,6)(7,9)(8,10)(11,12)(13,16,17,14)(15,19,20,18)(21,24,25,27)(22,23,26,28);(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,18)(17,20)(21,23)(22,24)(25,28)(26,27)Polycyclic presentation:
2^9|a^2=fg,b^2=ghi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1176650);