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 | |
1348 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)(11,12)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,32)(29,30)(33,34,38,40)(35,41,43,36)(37,39,45,46)(42,47,48,44);(1,2,4,6)(3,5,7,8)(10,14)(15,16)(18,22)(23,25)(24,30)(31,32)(37,45)(39,46)(42,48)(44,47);(1,2)(3,5)(4,6)(7,8)(9,10,13,14)(11,15,12,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,31)(29,30)(33,35,38,43)(34,36,40,41)(37,42,45,48)(39,44,46,47);(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,28)(21,27)(22,25)(24,32)(26,29)(30,31)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48);(1,3,4,7)(2,5,6,8)(9,12,13,11)(10,16,14,15)(17,20,21,29)(18,24,22,30)(19,26,27,28)(23,31,25,32)(33,37)(34,39)(35,42)(36,44)(38,45)(40,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=g,b^2=fhi,c^2=gi,d^2=i,e^2=h,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, 5815557);