Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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 | |
1826 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)(5,6,9,7)(8,11,12,10)(15,21)(16,23)(19,24)(20,25)(29,30,34,32)(31,36,39,35)(33,40,42,37)(38,43,41,44);(14,18)(19,24)(20,25)(26,28)(29,31,34,39)(30,35,32,36)(33,38,42,41)(37,43,40,44);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31)(30,36)(32,35)(33,41)(34,39)(37,43)(38,42)(40,44);(5,8)(6,10)(7,11)(9,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32,34,30)(31,35,39,36)(33,40,42,37)(38,43,41,44);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33,34,42)(30,37,32,40)(31,38,39,41)(35,43,36,44)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=i,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4295888);