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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1806 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)(5,6,8,9)(7,10,12,11)(13,14)(15,21)(16,23)(17,25)(18,19)(20,26)(22,24)(27,28)(30,34)(31,37)(36,40)(39,41);(6,9)(10,11)(14,19)(16,21)(17,26)(22,27)(29,30,33,34)(31,38,37,35)(32,40,42,36)(39,43,41,44);(5,6,8,9)(7,11,12,10)(13,15)(14,16)(17,27)(18,23)(19,21)(20,28)(22,26)(24,25)(29,31,33,37)(30,35,34,38)(32,41,42,39)(36,44,40,43);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,15)(17,24)(18,21)(19,23)(20,22)(25,27)(26,28)(29,30)(31,35)(32,36)(33,34)(37,38)(39,43)(40,42)(41,44);(5,7)(6,10)(8,12)(9,11)(13,17)(14,20)(15,22)(16,24)(18,26)(19,25)(21,28)(23,27)(29,32)(30,36)(31,39)(33,42)(34,40)(35,43)(37,41)(38,44)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=h,d^2=fi,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, 5187526);