Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | |
1774 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)(7,10)(8,12)(13,14,17,20)(15,18,21,25)(16,23,24,19)(22,28,27,26)(29,30)(31,36)(32,37)(33,41)(34,35)(38,43)(39,40)(42,44);(5,6,9,11)(7,8,10,12)(15,21)(18,25)(22,27)(26,28)(30,35)(31,37)(38,41)(39,44);(5,7,9,10)(6,8,11,12)(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,28)(24,27)(29,31,34,37)(30,36,35,32)(33,39,43,44)(38,42,41,40);(1,2)(3,4)(5,8)(6,10)(7,11)(9,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,24)(27,28)(29,32,34,36)(30,37,35,31)(33,42,43,40)(38,39,41,44);(13,16,17,24)(14,19,20,23)(15,22,21,27)(18,26,25,28)(29,33,34,43)(30,38,35,41)(31,39,37,44)(32,40,36,42)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=fg,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4327384);