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 | |
1412 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)(3,6,7,8)(9,10,12,14)(11,13,15,16)(18,22)(19,25)(20,27)(30,32)(33,34,38,37)(35,39,43,42)(36,44,46,40)(41,48,45,47);(3,7)(6,8)(9,11,12,15)(10,13,14,16)(19,25)(23,28)(26,31)(30,32)(34,37)(39,42)(40,44)(47,48);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,43)(34,39,37,42)(36,45,46,41)(40,48,44,47);(1,4,5,2)(3,8,7,6)(9,10)(11,13)(12,14)(15,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,36,38,46)(34,40,37,44)(35,41,43,45)(39,47,42,48);(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,34)(35,42,43,39)(36,44,46,40)(41,48,45,47)Polycyclic presentation:
2^9|a^2=fh,b^2=fgi,c^2=fg,d^2=hi,e^2=f,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, 1589284);