Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| 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:
(1,2,4,6)(3,5,7,8)(9,10,13,12)(11,14,15,16)(17,18,21,22)(19,25,27,23)(20,28,30,24)(26,31,29,32)(33,34)(35,42)(36,37)(38,40)(39,41)(43,46)(44,45)(47,48);(11,15)(14,16)(18,22)(23,25)(24,28)(31,32)(33,35,39,36)(34,37,41,42)(38,43,47,44)(40,45,48,46);(9,11)(10,14)(12,16)(13,15)(17,18)(19,23)(20,28)(21,22)(24,30)(25,27)(26,32)(29,31)(33,36,39,35)(34,37,41,42)(38,43,47,44)(40,46,48,45);(1,2)(3,5)(4,6)(7,8)(9,10,13,12)(11,14,15,16)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31)(33,37)(34,35)(36,41)(38,46)(39,42)(40,44)(43,48)(45,47);(1,3,4,7)(2,5,6,8)(9,12,13,10)(11,16,15,14)(17,20,21,30)(18,24,22,28)(19,26,27,29)(23,31,25,32)(33,38,39,47)(34,40,41,48)(35,43,36,44)(37,45,42,46)Polycyclic presentation:
2^9|a^2=fhi,b^2=fg,c^2=fg,d^2=gh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3562416);