Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1800 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)(5,6,8,9)(7,10,11,12)(15,21)(16,23)(19,24)(20,25)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(7,11)(10,12)(14,18)(19,24)(20,25)(26,28)(29,30,33,36)(31,37,39,34)(32,35,41,40)(38,43,44,42);(6,9)(10,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,30)(31,37)(32,40)(33,36)(34,39)(35,41)(38,42)(43,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(5,7)(6,10)(8,11)(9,12)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,32)(30,35)(31,38)(33,41)(34,42)(36,40)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fh,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5124849);