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 | |
1696 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)(5,6,9,10)(7,12,11,8)(14,18)(15,21)(20,25)(22,27)(29,30)(31,37)(32,34)(33,36)(35,38)(39,44)(40,42)(41,43);(6,10)(7,11)(15,21)(19,24)(22,27)(26,28)(30,32)(31,38)(36,40)(39,41);(6,10)(8,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31)(30,35)(32,37)(33,41)(34,38)(36,44)(39,42)(40,43);(1,2)(3,4)(5,7,9,11)(6,8,10,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32,34,30)(31,37,38,35)(33,40,42,36)(39,44,41,43);(5,8)(6,11)(7,10)(9,12)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33,34,42)(30,36,32,40)(31,39,38,41)(35,43,37,44)Polycyclic presentation:
2^9|a^2=fh,c^2=fg,d^2=f,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, 4511097);