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 | |
1698 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)(5,6,9,7)(8,10,12,11)(13,14,18,19)(15,23,26,20)(16,21,22,17)(24,25,28,27)(29,30)(31,35)(32,34)(33,40)(36,39)(37,42)(38,44)(41,43);(8,12)(10,11)(14,19)(17,21)(20,23)(25,27)(29,31,34,39)(30,35,32,36)(33,41,42,38)(37,44,40,43);(6,7)(10,11)(13,15,18,26)(14,20,19,23)(16,24,22,28)(17,25,21,27)(29,31)(30,36)(32,35)(33,38)(34,39)(37,44)(40,43)(41,42);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16)(14,21)(15,24)(17,19)(18,22)(20,27)(23,25)(26,28)(29,32,34,30)(31,36,39,35)(33,40,42,37)(38,44,41,43);(5,8,9,12)(6,10,7,11)(13,17,18,21)(14,22,19,16)(15,25,26,27)(20,28,23,24)(29,33)(30,37)(31,38)(32,40)(34,42)(35,43)(36,44)(39,41)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=f,d^2=fg,e^2=gh,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, 5533199);