Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(6,7)(8,12)(13,14,17,18)(15,21,22,19)(16,24,25,20)(23,26,28,27)(29,30,34,38)(31,36,32,35)(33,37,41,44)(39,43,40,42);(6,7)(10,11)(14,18)(15,22)(16,25)(26,27)(29,31,34,32)(30,35,38,36)(33,40,41,39)(37,43,44,42);(5,6,9,7)(8,10,12,11)(13,14)(15,21)(16,20)(17,18)(19,22)(23,27)(24,25)(26,28)(29,32,34,31)(30,36,38,35)(33,40,41,39)(37,43,44,42);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,19)(16,23)(17,22)(18,21)(20,26)(24,27)(25,28)(29,30)(31,36)(32,35)(33,37)(34,38)(39,43)(40,42)(41,44);(5,8)(6,10)(7,11)(9,12)(13,16,17,25)(14,20,18,24)(15,23,22,28)(19,26,21,27)(29,33,34,41)(30,37,38,44)(31,39,32,40)(35,42,36,43)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=fh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4374774);