Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5020720 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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:
(1,2,3,4)(6,9)(10,11)(14,18)(16,24)(19,21)(22,28)(31,37)(33,41)(35,38)(36,40);(13,14)(15,19)(16,20)(17,18)(21,23)(22,26)(24,25)(27,28)(29,30,34,32)(31,38,37,35)(33,36,41,40)(39,44,43,42);(5,6)(7,11)(8,9)(10,12)(13,14)(15,21)(16,25)(17,18)(19,23)(20,24)(22,26)(27,28)(29,31,34,37)(30,35,32,38)(33,39,41,43)(36,42,40,44);(1,2)(3,4)(5,7,8,12)(6,10,9,11)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27)(29,32,34,30)(31,35,37,38)(33,40,41,36)(39,42,43,44);(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27)(29,33)(30,36)(31,39)(32,40)(34,41)(35,42)(37,43)(38,44)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4928635);