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 | |
1718 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)(7,8)(11,12)(13,14,17,18)(15,19,21,25)(16,20,24,23)(22,26,27,28)(29,30)(31,38)(32,36)(33,37)(34,35)(39,44)(40,43)(41,42);(6,10)(11,12)(14,18)(15,21)(20,23)(22,27)(29,31,35,32)(30,36,34,38)(33,39,42,40)(37,43,41,44);(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,23)(17,18)(20,24)(21,25)(22,28)(26,27)(29,32,35,31)(30,36,34,38)(33,40,42,39)(37,43,41,44);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,15,17,21)(14,19,18,25)(16,22,24,27)(20,26,23,28)(29,33,35,42)(30,37,34,41)(31,39,32,40)(36,43,38,44);(5,8,9,7)(6,12,10,11)(13,16,17,24)(14,20,18,23)(15,22,21,27)(19,26,25,28)(29,34,35,30)(31,38,32,36)(33,41,42,37)(39,44,40,43)Polycyclic presentation:
2^9|a^2=hi,b^2=fg,c^2=fg,d^2=fgh,e^2=fgh,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, 4521914);