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 | |
1746 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)(7,11)(10,12)(13,14)(15,19)(16,20)(17,18)(21,26)(22,25)(23,24)(27,28)(30,34)(35,37)(36,40)(42,43);(6,9)(10,12)(14,18)(19,25)(20,23)(26,27)(29,30,33,34)(31,37,39,35)(32,36,41,40)(38,43,44,42);(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,23)(17,18)(20,24)(21,27)(22,25)(26,28)(29,30)(31,35)(32,40)(33,34)(36,41)(37,39)(38,43)(42,44);(1,2)(3,4)(13,15,17,22)(14,19,18,25)(16,21,24,28)(20,26,23,27)(29,31)(30,35)(32,38)(33,39)(34,37)(36,42)(40,43)(41,44);(5,7)(6,10)(8,11)(9,12)(13,16,17,24)(14,20,18,23)(15,21,22,28)(19,26,25,27)(29,32,33,41)(30,36,34,40)(31,38,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fh,d^2=fg,e^2=f,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, 4514864);