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 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,4)(3,7,5,8)(11,15)(13,16)(17,18,21,24)(19,25,26,22)(20,23,28,30)(27,31,32,29)(33,34,38,36)(35,41,43,39)(37,44,46,40)(42,47,45,48);(1,3,6,5)(2,7,4,8)(19,26)(22,25)(27,32)(29,31)(34,36)(39,41)(40,44)(47,48);(1,4,6,2)(3,7,5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)(33,35)(34,39)(36,41)(37,45)(38,43)(40,48)(42,46)(44,47);(1,3,6,5)(2,8,4,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,22)(20,27)(21,26)(23,29)(24,25)(28,32)(30,31)(33,36,38,34)(35,41,43,39)(37,44,46,40)(42,48,45,47);(1,5,6,3)(2,7,4,8)(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=fgi,d^2=fh,e^2=fgh,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, 4554776);