Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1734 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)(7,11)(8,10)(9,12)(16,23)(20,25)(21,27)(26,28)(29,30)(31,34)(32,37)(33,36)(35,39)(38,41)(40,43)(42,44);(5,7,10,9)(6,11,8,12)(14,18)(19,24)(20,25)(26,28)(30,31)(32,39)(36,38)(40,44);(5,6,10,8)(7,12,9,11)(13,14,17,18)(15,19,22,24)(16,20,23,25)(21,26,27,28)(29,31,34,30)(32,37,39,35)(33,36,41,38)(40,42,44,43);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(13,15,17,22)(14,19,18,24)(16,21,23,27)(20,26,25,28)(29,32)(30,35)(31,37)(33,40)(34,39)(36,42)(38,43)(41,44);(5,9,10,7)(6,11,8,12)(13,16)(14,20)(15,21)(17,23)(18,25)(19,26)(22,27)(24,28)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=h,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4517174);