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 | |
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)(3,8)(4,7)(5,6)(9,10,13,14)(11,12,15,16)(18,22)(19,25)(24,29)(27,31)(34,38)(35,41)(40,43)(42,47);(2,5)(3,4)(11,15)(12,16)(18,22)(19,25)(20,28)(30,32)(33,34,37,38)(35,39,41,45)(36,43,44,40)(42,48,47,46);(1,3,6,4)(2,7,5,8)(9,11)(10,12)(13,15)(14,16)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,34)(35,39)(36,43)(37,38)(40,44)(41,45)(42,48)(46,47);(1,4,6,3)(2,7,5,8)(9,12)(10,11)(13,16)(14,15)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,35,37,41)(34,39,38,45)(36,42,44,47)(40,46,43,48);(1,5,6,2)(3,7,4,8)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,36,37,44)(34,40,38,43)(35,42,41,47)(39,46,45,48)Polycyclic presentation:
2^9|a^2=gh,b^2=fhi,c^2=fghi,d^2=g,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, 6178244);