Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1610 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)(5,6,10,11)(7,12,9,8)(14,18)(15,21)(20,25)(22,27)(29,30,34,32)(31,37,39,35)(33,36,41,40)(38,43,44,42);(5,7,10,9)(6,8,11,12)(15,21)(19,24)(22,27)(26,28)(30,32)(35,37)(36,40)(42,43);(5,8)(6,7)(9,11)(10,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31)(30,35)(32,37)(33,38)(34,39)(36,42)(40,43)(41,44);(1,2)(3,4)(5,9,10,7)(6,8,11,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32,34,30)(31,37,39,35)(33,36,41,40)(38,42,44,43);(5,8)(6,9)(7,11)(10,12)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33,34,41)(30,36,32,40)(31,38,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fgh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4770569);