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 | #2416 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3937 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,5,10,9)(13,14,16,15)(17,18)(19,21)(20,24)(22,23)(25,26)(27,31)(28,29)(30,32)(33,34,39,38)(35,40,42,37)(36,41,45,46)(43,47,48,44);(2,7)(3,12)(4,10)(14,15)(17,19,22,26)(18,21,23,25)(20,28,30,27)(24,29,32,31)(33,35)(34,40)(36,43)(37,38)(39,42)(41,47)(44,46)(45,48);(2,7)(4,10)(8,11)(13,16)(18,23)(19,26)(24,32)(27,28)(35,42)(37,40)(43,48)(44,47);(1,3,6,12)(2,8,7,11)(4,13,10,16)(5,15,9,14)(17,20,22,30)(18,24,23,32)(19,27,26,28)(21,31,25,29)(33,36)(34,41)(35,43)(37,44)(38,46)(39,45)(40,47)(42,48);(1,4,6,10)(2,9,7,5)(3,13,12,16)(8,14,11,15)(17,19)(18,25)(20,27)(21,23)(22,26)(24,29)(28,30)(31,32)(33,37)(34,35)(36,44)(38,42)(39,40)(41,43)(45,47)(46,48);(1,5,6,9)(2,10,7,4)(3,14,12,15)(8,13,11,16)(17,21,22,25)(18,19,23,26)(20,29,30,31)(24,28,32,27)(33,38,39,34)(35,37,42,40)(36,41,45,46)(43,47,48,44)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8541545);