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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1895 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)(7,11)(8,10)(9,12)(13,14)(15,21)(16,22)(17,20)(18,19)(23,26)(24,28)(25,27)(31,37)(32,39)(34,40)(35,42);(5,6)(7,11)(8,10)(9,12)(14,19)(15,22)(17,25)(24,26)(29,30,33,36)(31,34,37,40)(32,35,39,42)(38,41,43,44);(5,6,8,10)(7,9,12,11)(13,15)(14,16)(17,23)(18,22)(19,21)(20,24)(25,28)(26,27)(29,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,44);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16)(14,15)(17,26)(18,21)(19,22)(20,28)(23,27)(24,25)(29,31,33,37)(30,34,36,40)(32,38,39,43)(35,41,42,44);(5,7,8,12)(6,9,10,11)(13,17,18,25)(14,20,19,27)(15,23,22,28)(16,24,21,26)(29,32)(30,35)(31,38)(33,39)(34,41)(36,42)(37,43)(40,44)Polycyclic presentation:
2^9|b^2=gh,c^2=fg,d^2=gh,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, 5524135);