Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1762 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)(7,11)(9,12)(13,14)(15,21)(16,18)(17,20)(19,23)(22,27)(24,25)(26,28)(29,30,34,32)(31,37,39,35)(33,36,42,40)(38,44,41,43);(5,6,8,10)(7,9,11,12)(14,16)(19,21)(20,24)(26,27)(29,31)(30,35)(32,37)(33,38)(34,39)(36,43)(40,44)(41,42);(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,21)(17,22)(18,23)(20,26)(24,27)(25,28)(29,32,34,30)(31,37,39,35)(33,40,42,36)(38,44,41,43);(1,2)(3,4)(13,16,18,14)(15,21,23,19)(17,20,25,24)(22,26,28,27)(29,31)(30,35)(32,37)(33,41)(34,39)(36,44)(38,42)(40,43);(5,7)(6,9)(8,11)(10,12)(13,17,18,25)(14,20,16,24)(15,22,23,28)(19,26,21,27)(29,33,34,42)(30,36,32,40)(31,38,39,41)(35,43,37,44)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=fg,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^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4769361);