Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1872 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)(5,6,8,10)(7,9,11,12)(13,14)(15,21)(16,18)(17,24)(19,23)(20,25)(22,27)(26,28)(29,30,33,36)(31,37,39,34)(32,35,40,42)(38,43,44,41);(5,7,8,11)(6,9,10,12)(14,16)(17,25)(19,21)(22,26)(32,40)(35,42)(38,44)(41,43);(5,7)(6,9)(8,11)(10,12)(13,15)(14,19)(16,21)(17,26)(18,23)(20,28)(22,25)(24,27)(29,31,33,39)(30,34,36,37)(32,38,40,44)(35,41,42,43);(1,2)(3,4)(13,16,18,14)(15,21,23,19)(17,24,25,20)(22,28,26,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,42)(41,44);(5,6)(7,9)(8,10)(11,12)(13,17)(14,20)(15,22)(16,24)(18,25)(19,27)(21,28)(23,26)(29,32,33,40)(30,35,36,42)(31,38,39,44)(34,41,37,43)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=fg,d^2=f,e^2=fg,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, 5120528);