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 | |
1850 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)(13,14,18,16)(15,21,23,19)(17,20,25,24)(22,27,28,26)(29,30,33,37)(31,35,39,34)(32,36,41,40)(38,43,44,42);(5,6)(7,10)(8,9)(11,12)(17,25)(20,24)(22,28)(26,27)(29,31,33,39)(30,34,37,35)(32,38,41,44)(36,42,40,43);(7,11)(10,12)(13,15,18,23)(14,19,16,21)(17,22,25,28)(20,26,24,27)(29,31)(30,35)(32,38)(33,39)(34,37)(36,43)(40,42)(41,44);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16,18,14)(15,19,23,21)(17,24,25,20)(22,26,28,27)(29,30)(31,34)(32,40)(33,37)(35,39)(36,41)(38,43)(42,44);(5,6)(7,12)(8,9)(10,11)(13,17,18,25)(14,20,16,24)(15,22,23,28)(19,26,21,27)(29,32)(30,36)(31,38)(33,41)(34,42)(35,43)(37,40)(39,44)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,d^2=fgh,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, 5525131);