Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(6,9)(7,11)(14,18)(19,24)(20,25)(26,28)(29,30)(31,36)(32,35)(33,37)(34,39)(38,43)(40,42)(41,44);(5,6)(7,10)(8,9)(11,12)(15,21)(19,24)(22,27)(26,28)(29,31,35,34)(30,36,32,39)(33,41,42,38)(37,44,40,43);(7,11)(10,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,32,35,30)(31,36,34,39)(33,40,42,37)(38,43,41,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,33,35,42)(30,37,32,40)(31,38,34,41)(36,43,39,44);(5,7,8,11)(6,10,9,12)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,34,35,31)(30,36,32,39)(33,41,42,38)(37,43,40,44)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=h,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4515629);