Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(14,18)(15,21)(16,23)(26,28)(29,30)(31,35)(32,39)(33,34)(36,41)(37,44)(38,42)(40,43);(5,7,9,10)(6,8,11,12)(14,18)(15,21)(20,25)(22,27)(30,34)(35,42)(36,39)(43,44);(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,30)(31,35)(32,36)(33,34)(37,43)(38,42)(39,41)(40,44);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,31,33,38)(30,35,34,42)(32,40,41,37)(36,44,39,43);(5,8)(6,10)(7,11)(9,12)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,32)(30,36)(31,37)(33,41)(34,39)(35,43)(38,40)(42,44)Polycyclic presentation:
2^9|a^2=gi,b^2=fgh,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4775897);