Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1743 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)(9,12)(13,14)(15,18)(16,21)(17,22)(19,23)(20,25)(24,26)(27,28)(29,30,33,36)(31,37,39,34)(32,35,40,42)(38,43,44,41);(5,6,8,10)(7,9,11,12)(14,15)(16,23)(17,25)(26,27)(32,40)(35,42)(38,44)(41,43);(5,6)(7,9)(8,10)(11,12)(13,15,18,14)(16,21,23,19)(17,20,25,22)(24,26,28,27)(29,31,33,39)(30,34,36,37)(32,38,40,44)(35,41,42,43);(1,2,3,4)(13,16)(14,19)(15,21)(17,24)(18,23)(20,26)(22,27)(25,28)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,42)(41,44);(5,7,8,11)(6,9,10,12)(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)(29,32,33,40)(30,35,36,42)(31,38,39,44)(34,41,37,43)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=g,d^2=gi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 4990758);