Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,7)(8,11,12,10)(13,14,17,20)(15,18,21,25)(16,23,24,19)(22,28,27,26)(29,30)(31,36)(32,40)(33,38)(34,35)(37,42)(39,43)(41,44);(2,4)(5,7,9,6)(8,11,12,10)(15,21)(18,25)(22,27)(26,28)(30,35)(31,38)(37,40)(39,44);(1,2)(3,4)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,24)(27,28)(29,31,34,38)(30,36,35,33)(32,39,42,44)(37,41,40,43);(5,8,9,12)(6,10,7,11)(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,28)(24,27)(29,32,34,42)(30,37,35,40)(31,39,38,44)(33,43,36,41);(13,16,17,24)(14,19,20,23)(15,22,21,27)(18,26,25,28)(29,33,34,36)(30,38,35,31)(32,41,42,43)(37,39,40,44)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,c^2=f,d^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2565243);