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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1803 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)(6,10)(8,11)(16,23)(20,25)(21,27)(26,28)(29,30)(31,37)(32,40)(33,36)(34,39)(35,42)(38,43)(41,44);(6,10)(7,12)(14,18)(19,24)(20,25)(26,28)(29,30,33,36)(31,34,39,37)(32,35,42,40)(38,43,41,44);(5,6)(7,11)(8,12)(9,10)(13,14,17,18)(15,19,22,24)(16,20,23,25)(21,26,27,28)(29,30)(31,34)(32,35)(33,36)(37,39)(38,43)(40,42)(41,44);(1,2)(3,4)(5,7,9,12)(6,11,10,8)(13,15,17,22)(14,19,18,24)(16,21,23,27)(20,26,25,28)(29,31,33,39)(30,34,36,37)(32,41,42,38)(35,44,40,43);(5,8,9,11)(6,12,10,7)(13,16)(14,20)(15,21)(17,23)(18,25)(19,26)(22,27)(24,28)(29,32,33,42)(30,35,36,40)(31,38,39,41)(34,43,37,44)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=fgh,d^2=h,e^2=fg,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, 4776025);