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)(13,14,17,18)(15,21,23,19)(16,20,24,26)(22,28,25,27)(29,30,34,37)(31,35,32,39)(33,36,41,44)(38,42,40,43);(5,6)(7,10)(8,9)(11,12)(14,18)(16,24)(19,21)(22,25)(29,31,34,32)(30,35,37,39)(33,40,41,38)(36,43,44,42);(7,11)(10,12)(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,26)(25,28)(29,32,34,31)(30,35,37,39)(33,40,41,38)(36,42,44,43);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,17,23)(14,19,18,21)(16,25,24,22)(20,28,26,27)(29,30)(31,35)(32,39)(33,36)(34,37)(38,42)(40,43)(41,44);(5,6)(7,12)(8,9)(10,11)(13,16,17,24)(14,20,18,26)(15,22,23,25)(19,27,21,28)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,44)(39,43)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gh,d^2=h,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, 5543875);