Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1294 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)(12,16)(14,15)(20,27)(24,29)(25,31)(30,32)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(2,5)(7,8)(9,10,13,11)(12,14,16,15)(18,22)(23,28)(24,29)(30,32)(36,44)(39,46)(42,48)(45,47);(1,3,6,4)(2,7,5,8)(9,11,13,10)(12,15,16,14)(17,18,21,22)(19,23,26,28)(20,24,27,29)(25,30,31,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,4,6,3)(2,7,5,8)(9,12)(10,14)(11,15)(13,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,5,6,2)(3,8,4,7)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=fhi,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5667792);