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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(10,12)(13,14)(15,21)(16,23)(17,25)(18,19)(20,27)(22,24)(26,28)(29,30,35,38)(31,39,32,36)(33,34,44,37)(40,43,42,41);(6,9)(10,12)(14,19)(16,21)(20,25)(24,28)(29,31,35,32)(30,36,38,39)(33,40,44,42)(34,41,37,43);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,26)(18,23)(19,21)(20,28)(22,27)(24,25)(29,32,35,31)(30,36,38,39)(33,42,44,40)(34,41,37,43);(1,2,3,4)(13,16)(14,15)(17,24)(18,21)(19,23)(20,22)(25,26)(27,28)(29,33)(30,37)(31,40)(32,42)(34,38)(35,44)(36,43)(39,41);(5,7,8,11)(6,10,9,12)(13,17,18,27)(14,20,19,25)(15,22,23,26)(16,24,21,28)(29,34,35,37)(30,33,38,44)(31,41,32,43)(36,40,39,42)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fg,d^2=fi,e^2=fh,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, 4576027);