Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(11,15)(13,16)(18,22)(19,25)(20,27)(30,32)(33,34,38,39)(35,36,40,44)(37,41,45,46)(42,43,47,48);(1,2,5,4)(3,7,8,6)(9,10,12,14)(11,13,15,16)(18,22)(19,25)(24,29)(26,31)(34,39)(35,40)(37,45)(43,48);(1,3,5,8)(2,6,4,7)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35)(34,36)(37,42)(38,40)(39,44)(41,43)(45,47)(46,48);(9,11,12,15)(10,13,14,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,36)(34,40)(35,39)(37,43)(38,44)(41,47)(42,46)(45,48);(1,4,5,2)(3,7,8,6)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37)(34,41)(35,42)(36,43)(38,45)(39,46)(40,47)(44,48)Polycyclic presentation:
2^9|a^2=f,b^2=fhi,c^2=fgh,d^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1458980);