Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
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:
(2,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,20)(19,25,27,23)(21,28,30,24)(26,31,29,32)(33,34,38,37)(35,41,43,39)(36,40,44,45)(42,47,48,46);(9,10,12,14)(11,13,16,15)(18,20)(23,25)(24,28)(31,32)(36,44)(40,45)(42,48)(46,47);(9,10)(11,15)(12,14)(13,16)(17,19,22,27)(18,23,20,25)(21,29,30,26)(24,32,28,31)(33,35,38,43)(34,39,37,41)(36,42,44,48)(40,46,45,47);(1,2)(3,6)(4,5)(7,8)(17,20,22,18)(19,25,27,23)(21,28,30,24)(26,32,29,31)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,13,14,15)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,37,38,34)(35,41,43,39)(36,45,44,40)(42,47,48,46)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=fh,e^2=ghi,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, 4664708);