Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1845 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)(5,6,10,8)(7,12,9,11)(13,14)(15,21)(16,18)(17,24)(19,23)(20,25)(22,27)(26,28)(29,30,33,34)(31,37,38,35)(32,36,40,42)(39,44,41,43);(5,6,10,8)(7,11,9,12)(14,16)(17,25)(19,21)(22,26)(30,34)(31,38)(32,40)(43,44);(5,7,10,9)(6,11,8,12)(13,15)(14,19)(16,21)(17,26)(18,23)(20,28)(22,25)(24,27)(29,31,33,38)(30,35,34,37)(32,41,40,39)(36,44,42,43);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(13,16,18,14)(15,21,23,19)(17,24,25,20)(22,28,26,27)(29,30)(31,35)(32,36)(33,34)(37,38)(39,43)(40,42)(41,44);(5,9,10,7)(6,11,8,12)(13,17)(14,20)(15,22)(16,24)(18,25)(19,27)(21,28)(23,26)(29,32,33,40)(30,36,34,42)(31,39,38,41)(35,43,37,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=fgh,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5135042);