Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1594 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)(6,10)(7,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,27)(25,26)(29,30,33,36)(31,37,39,34)(32,35,40,42)(38,43,44,41);(5,6,9,10)(7,11,12,8)(14,18)(19,21)(20,26)(27,28)(32,40)(35,42)(38,44)(41,43);(5,7,9,12)(6,11,10,8)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,27)(25,26)(29,31,33,39)(30,34,36,37)(32,38,40,44)(35,41,42,43);(1,2)(3,4)(5,8)(6,7)(9,11)(10,12)(13,15,17,23)(14,19,18,21)(16,24,25,22)(20,28,26,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,42)(41,44);(5,6)(7,11)(8,12)(9,10)(13,16,17,25)(14,20,18,26)(15,22,23,24)(19,27,21,28)(29,32,33,40)(30,35,36,42)(31,38,39,44)(34,41,37,43)Polycyclic presentation:
2^9|a^2=fgh,b^2=fh,c^2=g,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5524886);