Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
1292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,6,8,4)(10,15)(11,12)(20,27)(24,29)(25,31)(30,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,46)(45,47);(1,3,5,8)(2,4,7,6)(9,10,14,15)(11,13,12,16)(18,22)(23,28)(24,29)(30,32)(34,38)(36,44)(39,41)(42,45);(1,3,5,8)(2,6,7,4)(9,11,14,12)(10,16,15,13)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,35)(34,39)(36,45)(37,43)(38,41)(40,48)(42,44)(46,47);(1,3)(2,6)(4,7)(5,8)(9,12,14,11)(10,16,15,13)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48);(1,4,5,6)(2,3,7,8)(9,13,14,16)(10,11,15,12)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,36,37,44)(34,40,38,46)(35,42,43,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=fg,b^2=fgi,c^2=fgi,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6178684);