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 | |
1296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,8,7)(11,15)(14,16)(18,22)(19,25)(24,29)(27,31)(33,34,38,42)(35,40,36,39)(37,45,46,41)(43,47,44,48);(1,2)(3,5)(4,6)(7,8)(10,13)(11,15)(18,22)(19,25)(20,28)(30,32)(33,35,38,36)(34,39,42,40)(37,44,46,43)(41,48,45,47);(1,2,4,6)(3,7,8,5)(9,10,12,13)(11,14,15,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36,38,35)(34,40,42,39)(37,43,46,44)(41,47,45,48);(1,2)(3,5)(4,6)(7,8)(9,11,12,15)(10,14,13,16)(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);(1,3,4,8)(2,5,6,7)(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=f,b^2=hi,c^2=gh,d^2=fgh,e^2=fghi,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, 6247234);