Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | #16291 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2254 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)(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,17)(18,20)(21,22,24,27)(23,25,28,26);(7,11)(10,12)(13,15,17,20)(14,18,16,19)(21,23)(22,25)(24,28)(26,27);(6,9)(10,12)(14,16)(18,19)(21,23,24,28)(22,26,27,25);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,18)(17,20)(21,23)(22,26)(24,28)(25,27);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,18,20,19)(21,22)(23,26)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=h,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5129454);