Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1959 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)(5,6,8,10)(7,11,12,9)(13,14)(15,21)(16,18)(17,20)(19,23)(22,27)(24,25)(26,28)(30,34)(31,37)(32,40)(43,44);(7,12)(9,11)(14,16)(19,21)(20,24)(26,27)(29,30,33,34)(31,38,37,35)(32,36,40,42)(39,44,41,43);(5,6)(7,9)(8,10)(11,12)(13,15,18,23)(14,19,16,21)(17,22,25,28)(20,26,24,27)(29,30)(31,35)(32,36)(33,34)(37,38)(39,43)(40,42)(41,44);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,16,18,14)(15,21,23,19)(17,20,25,24)(22,26,28,27)(29,31,33,37)(30,35,34,38)(32,41,40,39)(36,44,42,43);(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)(29,32,33,40)(30,36,34,42)(31,39,37,41)(35,43,38,44)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=fh,d^2=g,e^2=h,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, 4773680);