Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1308 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)(3,7)(4,5)(6,8)(9,10,13,12)(11,15,16,14)(18,22)(23,26)(24,29)(30,32)(33,34,38,42)(35,40,36,39)(37,45,46,41)(43,47,44,48);(1,3)(2,6)(4,7)(5,8)(10,12)(11,16)(19,25)(23,26)(27,31)(30,32)(33,35,38,36)(34,39,42,40)(37,44,46,43)(41,48,45,47);(3,8)(6,7)(9,10,13,12)(11,15,16,14)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,36,38,35)(34,40,42,39)(37,44,46,43)(41,48,45,47);(1,3)(2,7)(4,6)(5,8)(9,11,13,16)(10,14,12,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,48)(45,46);(1,4,5,2)(3,6,8,7)(9,12,13,10)(11,15,16,14)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,37,38,46)(34,41,42,45)(35,43,36,44)(39,47,40,48)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=fghi,d^2=fhi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6038141);