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 | |
1362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10,13,12)(11,14,16,15)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,34)(35,41)(36,38)(37,44)(39,42)(40,45)(43,46)(47,48);(1,2,5,4)(3,6,8,7)(9,11,13,16)(10,14,12,15)(18,19)(21,29)(23,25)(27,32)(34,36)(35,42)(37,45)(46,47);(1,2,5,4)(3,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31)(33,35)(34,39)(36,41)(37,43)(38,42)(40,46)(44,47)(45,48);(1,3,5,8)(2,6,4,7)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36,38,34)(35,39,42,41)(37,44,45,40)(43,46,48,47);(1,4,5,2)(3,7,8,6)(9,12,13,10)(11,15,16,14)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=hi,b^2=fghi,c^2=fi,d^2=fgi,e^2=fghi,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, 1872056);