Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(13,14,17,18)(15,19,21,22)(16,24,25,20)(23,28,27,26)(29,30)(31,34)(32,40)(33,36)(35,41)(37,39)(38,43)(42,44);(6,9)(10,12)(14,18)(15,21)(20,24)(23,27)(29,30,33,36)(31,34,39,37)(32,35,41,40)(38,42,44,43);(5,6,8,9)(7,10,11,12)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,25)(26,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,28)(25,27)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(5,7,8,11)(6,10,9,12)(13,16,17,25)(14,20,18,24)(15,23,21,27)(19,26,22,28)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=fh,d^2=fg,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 452041);