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 | |
1406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(5,8)(11,15)(14,16)(17,18,21,22)(19,23,25,29)(20,24,28,27)(26,30,31,32)(33,34,38,40)(35,41,42,36)(37,45,46,39)(43,44,48,47);(1,2,4,6)(3,5,7,8)(10,13)(14,16)(18,22)(19,25)(24,27)(26,31)(35,42)(36,41)(43,48)(44,47);(1,2)(3,5)(4,6)(7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31)(33,35)(34,36)(37,43)(38,42)(39,44)(40,41)(45,47)(46,48);(9,11)(10,14)(12,15)(13,16)(17,19,21,25)(18,23,22,29)(20,26,28,31)(24,30,27,32)(33,36)(34,35)(37,44)(38,41)(39,43)(40,42)(45,48)(46,47);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,28)(18,24,22,27)(19,26,25,31)(23,30,29,32)(33,37,38,46)(34,39,40,45)(35,43,42,48)(36,44,41,47)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=i,d^2=f,e^2=g,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, 4556436);