Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1698 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)(5,6,9,11)(7,12,10,8)(14,18)(15,21)(20,25)(22,27)(29,30,34,32)(31,35,37,38)(33,40,41,36)(39,44,43,42);(5,7,9,10)(6,8,11,12)(15,21)(19,24)(22,27)(26,28)(31,37)(35,38)(39,43)(42,44);(5,8,9,12)(6,10,11,7)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31)(30,35)(32,38)(33,39)(34,37)(36,42)(40,44)(41,43);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32,34,30)(31,38,37,35)(33,40,41,36)(39,44,43,42);(5,7)(6,8)(9,10)(11,12)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33)(30,36)(31,39)(32,40)(34,41)(35,42)(37,43)(38,44)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4375654);