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 | |
1922 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)(5,6,9,8)(7,11,12,10)(13,14,17,20)(15,18,21,25)(16,23,24,19)(22,28,27,26)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(5,6)(7,11)(8,9)(10,12)(15,21)(18,25)(22,27)(26,28)(29,30,33,36)(31,37,39,34)(32,35,41,40)(38,43,44,42);(6,8)(10,11)(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,28)(24,27)(29,30)(31,37)(32,40)(33,36)(34,39)(35,41)(38,42)(43,44);(1,2,3,4)(5,7,9,12)(6,10,8,11)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,24)(27,28)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(5,8,9,6)(7,11,12,10)(13,16,17,24)(14,19,20,23)(15,22,21,27)(18,26,25,28)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,d^2=fhi,e^2=h,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, 5191622);