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) = C26×C4 |
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 | |
1758 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)(5,6,9,12)(7,11,8,10)(13,14)(15,21)(16,18)(17,20)(19,23)(22,27)(24,25)(26,28)(30,34)(31,37)(32,40)(42,44);(7,8)(10,11)(14,16)(17,25)(19,21)(22,28)(29,30,33,34)(31,38,37,35)(32,41,40,36)(39,42,43,44);(5,7,9,8)(6,10,12,11)(13,15,18,23)(14,19,16,21)(17,22,25,28)(20,26,24,27)(29,30)(31,38)(32,36)(33,34)(35,37)(39,44)(40,41)(42,43);(1,2)(3,4)(5,8,9,7)(6,11,12,10)(13,16,18,14)(15,19,23,21)(17,24,25,20)(22,26,28,27)(29,31,33,37)(30,35,34,38)(32,39,40,43)(36,42,41,44);(5,6)(7,10)(8,11)(9,12)(13,17,18,25)(14,20,16,24)(15,22,23,28)(19,26,21,27)(29,32,33,40)(30,36,34,41)(31,39,37,43)(35,42,38,44)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=fh,d^2=g,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, 4374789);