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 | |
1316 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)(6,8)(9,10,13,11)(12,15,16,14)(17,18,21,22)(19,25,27,23)(20,24,28,30)(26,32,29,31)(33,34,38,41)(35,36,43,39)(37,40,45,47)(42,44,48,46);(2,5)(6,8)(9,10,13,11)(12,14,16,15)(18,22)(20,28)(23,25)(26,29)(37,45)(40,47)(42,48)(44,46);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,15,16,14)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,35,38,43)(34,39,41,36)(37,42,45,48)(40,46,47,44);(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,28)(30,31)(33,36,38,39)(34,35,41,43)(37,44,45,46)(40,42,47,48);(1,3,4,7)(2,6,5,8)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,32)(27,29)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=hi,d^2=fg,e^2=fh,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, 5708539);