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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1689 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,8)(11,12)(13,14)(15,18)(16,21)(17,20)(19,23)(22,25)(24,27)(26,28)(29,30,34,32)(31,37,39,35)(33,36,41,40)(38,43,44,42);(6,10)(11,12)(14,15)(16,23)(17,25)(26,27)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(5,6,9,10)(7,11,8,12)(13,15,18,14)(16,21,23,19)(17,20,25,22)(24,26,28,27)(29,32,34,30)(31,35,39,37)(33,36,41,40)(38,43,44,42);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,16)(14,19)(15,21)(17,24)(18,23)(20,26)(22,27)(25,28)(29,31)(30,35)(32,37)(33,38)(34,39)(36,42)(40,43)(41,44);(5,8,9,7)(6,12,10,11)(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=h,d^2=fh,e^2=fgh,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, 5133638);