Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3811 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,5)(3,8,11,7)(4,12,13,9)(10,14,16,15)(17,18,23,27)(19,24,22,26)(20,25,31,21)(28,30,29,32)(33,34,39,37)(35,40,38,46)(36,43,45,41)(42,48,44,47);(1,3,6,11)(2,7,5,8)(4,10,13,16)(9,14,12,15)(17,19,23,22)(18,24,27,26)(20,29,31,28)(21,30,25,32)(33,35,39,38)(34,40,37,46)(36,42,45,44)(41,47,43,48);(1,3,6,11)(2,7,5,8)(4,10,13,16)(9,14,12,15)(17,20,23,31)(18,25,27,21)(19,28,22,29)(24,30,26,32)(33,36)(34,41)(35,42)(37,43)(38,44)(39,45)(40,47)(46,48);(1,3)(2,8)(4,10)(5,7)(6,11)(9,15)(12,14)(13,16)(17,20)(18,21)(19,29)(22,28)(23,31)(24,30)(25,27)(26,32)(33,37,39,34)(35,40,38,46)(36,43,45,41)(42,47,44,48);(1,4,6,13)(2,9,5,12)(3,10,11,16)(7,14,8,15)(17,21)(18,20)(19,30)(22,32)(23,25)(24,29)(26,28)(27,31)(33,38,39,35)(34,40,37,46)(36,44,45,42)(41,47,43,48);(1,5,6,2)(3,8,11,7)(4,9,13,12)(10,14,16,15)(17,22,23,19)(18,26,27,24)(20,29,31,28)(21,30,25,32)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=gi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8657402);