Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1318 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)(3,8)(4,7)(5,6)(11,15)(13,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34,37,38)(35,41,43,39)(36,44,46,40)(42,47,45,48);(1,3,6,4)(2,7,5,8)(18,22)(23,25)(24,28)(30,31)(34,38)(39,41)(40,44)(47,48);(1,3,6,4)(2,8,5,7)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,31)(30,32)(33,35)(34,39)(36,45)(37,43)(38,41)(40,48)(42,46)(44,47);(1,4,6,3)(2,8,5,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48);(1,5,6,2)(3,8,4,7)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,36)(34,40)(35,42)(37,46)(38,44)(39,47)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=ghi,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4546359);