Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1667 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)(16,23)(17,20)(18,19)(22,28)(24,26)(25,27)(29,30,34,37)(31,35,32,39)(33,36,41,44)(38,42,40,43);(5,6)(7,11)(8,10)(9,12)(14,19)(16,21)(17,25)(22,26)(29,31,34,32)(30,35,37,39)(33,40,41,38)(36,43,44,42);(5,6,8,10)(7,9,12,11)(13,15)(14,16)(17,26)(18,23)(19,21)(20,28)(22,25)(24,27)(29,30)(31,35)(32,39)(33,36)(34,37)(38,42)(40,43)(41,44);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,16)(14,15)(17,24)(18,21)(19,23)(20,22)(25,28)(26,27)(29,32,34,31)(30,35,37,39)(33,40,41,38)(36,42,44,43);(5,7)(6,9)(8,12)(10,11)(13,17,18,25)(14,20,19,27)(15,22,23,26)(16,24,21,28)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,44)(39,43)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgh,c^2=h,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 4989646);