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 | |
1324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,4,7,8)(11,15)(13,16)(17,18,21,24)(19,25,26,22)(20,23,28,30)(27,31,32,29)(33,34,38,36)(35,39,43,41)(37,44,45,40)(42,47,48,46);(2,6)(3,7)(19,26)(22,25)(27,32)(29,31)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,3)(2,4)(5,7)(6,8)(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)(37,42)(38,43)(40,46)(44,47)(45,48);(9,11,12,15)(10,13,14,16)(17,19,21,26)(18,22,24,25)(20,27,28,32)(23,29,30,31)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,4)(2,7)(3,6)(5,8)(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,c^2=gi,d^2=fhi,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, 4610694);