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 | |
1596 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)(5,6,10,8)(7,11,9,12)(13,14,18,15)(16,21,24,19)(17,20,26,22)(23,28,25,27)(29,30,33,36)(31,37,39,34)(32,35,40,42)(38,43,44,41);(5,7,10,9)(6,11,8,12)(14,15)(19,21)(20,22)(27,28)(32,40)(35,42)(38,44)(41,43);(5,8,10,6)(7,11,9,12)(13,15,18,14)(16,19,24,21)(17,22,26,20)(23,27,25,28)(29,31,33,39)(30,34,36,37)(32,38,40,44)(35,41,42,43);(1,2)(3,4)(13,16,18,24)(14,19,15,21)(17,25,26,23)(20,28,22,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,42)(41,44);(5,9,10,7)(6,11,8,12)(13,17,18,26)(14,20,15,22)(16,23,24,25)(19,27,21,28)(29,32)(30,35)(31,38)(33,40)(34,41)(36,42)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,c^2=gh,d^2=f,e^2=h,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, 5536653);