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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1838 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)(7,9)(8,10)(11,12)(13,14,17,18)(15,21,23,19)(16,24,25,20)(22,26,28,27)(29,30,34,38)(31,35,36,32)(33,37,42,44)(39,43,41,40);(8,12)(10,11)(14,18)(16,25)(19,21)(22,28)(29,31,34,36)(30,32,38,35)(33,39,42,41)(37,40,44,43);(6,7)(10,11)(13,15,17,23)(14,19,18,21)(16,22,25,28)(20,26,24,27)(29,31)(30,35)(32,38)(33,41)(34,36)(37,40)(39,42)(43,44);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16,17,25)(14,20,18,24)(15,22,23,28)(19,26,21,27)(29,32,34,35)(30,36,38,31)(33,40,42,43)(37,41,44,39);(5,8)(6,10)(7,11)(9,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,33)(30,37)(31,39)(32,40)(34,42)(35,43)(36,41)(38,44)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fh,d^2=i,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, 5187424);