Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,8,10)(7,11,12,9)(15,21)(16,23)(19,24)(20,25)(29,30,34,32)(31,37,39,35)(33,36,42,40)(38,44,41,43);(14,18)(19,24)(20,25)(26,28)(29,31,34,39)(30,35,32,37)(33,41,42,38)(36,44,40,43);(5,7,8,12)(6,9,10,11)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31)(30,35)(32,37)(33,38)(34,39)(36,43)(40,44)(41,42);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32,34,30)(31,37,39,35)(33,40,42,36)(38,44,41,43);(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,42)(30,36,32,40)(31,38,39,41)(35,43,37,44)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4841886);