Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(7,11)(10,12)(14,18)(15,21)(16,23)(26,28)(29,30,34,35)(31,32,36,39)(33,37,43,44)(38,40,41,42);(1,2,3,4)(15,21)(19,24)(22,27)(26,28)(30,35)(32,39)(37,44)(40,42);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31,34,36)(30,32,35,39)(33,41,43,38)(37,42,44,40);(5,7,9,11)(6,10,8,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32)(30,36)(31,35)(33,42)(34,39)(37,38)(40,43)(41,44);(5,8,9,6)(7,12,11,10)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,43)(30,37,35,44)(31,38,36,41)(32,40,39,42)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fi,d^2=i,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3702270);