Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1898 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)(6,9)(7,11)(13,14,17,18)(15,21,23,19)(16,20,25,26)(22,28,24,27)(29,30,33,36)(31,37,38,34)(32,40,41,35)(39,42,44,43);(14,18)(19,21)(20,26)(27,28)(31,38)(34,37)(39,44)(42,43);(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,24)(17,23)(18,21)(20,28)(22,25)(26,27)(29,31,33,38)(30,34,36,37)(32,39,41,44)(35,42,40,43);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,28)(25,26)(29,30)(31,34)(32,35)(33,36)(37,38)(39,42)(40,41)(43,44);(13,16,17,25)(14,20,18,26)(15,22,23,24)(19,27,21,28)(29,32,33,41)(30,35,36,40)(31,39,38,44)(34,42,37,43)Polycyclic presentation:
2^9|a^2=fh,c^2=fg,d^2=gi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4297043);