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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1846 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)(6,9)(7,11)(14,18)(15,21)(16,24)(26,28)(29,30,34,32)(31,35,39,37)(33,40,41,36)(38,43,44,42);(5,6)(7,10)(8,9)(11,12)(14,18)(16,24)(19,22)(23,27)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(7,11)(10,12)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,25)(26,27)(29,31)(30,35)(32,37)(33,38)(34,39)(36,42)(40,43)(41,44);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,17,21)(14,19,18,22)(16,23,24,27)(20,26,25,28)(29,32,34,30)(31,37,39,35)(33,40,41,36)(38,43,44,42);(5,6)(7,12)(8,9)(10,11)(13,16,17,24)(14,20,18,25)(15,23,21,27)(19,26,22,28)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=h,e^2=g,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, 5524803);