Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(14,18)(16,23)(19,24)(22,27)(29,30,35,32)(31,37,33,36)(34,38,42,40)(39,44,41,43);(6,9)(10,12)(14,18)(15,21)(20,25)(22,27)(29,31,35,33)(30,36,32,37)(34,41,42,39)(38,44,40,43);(5,6,8,9)(7,10,11,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,32,35,30)(31,36,33,37)(34,40,42,38)(39,43,41,44);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,33,35,31)(30,37,32,36)(34,41,42,39)(38,44,40,43);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,34,35,42)(30,38,32,40)(31,39,33,41)(36,43,37,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1682236);