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 | #46787 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4191 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,10)(4,12)(5,9)(6,7)(8,13)(11,15)(14,16)(33,34)(35,40)(36,42)(37,38)(39,44)(41,45)(43,48)(46,47);(1,3,7,9)(2,5,6,10)(4,11,13,16)(8,15,12,14)(17,18,22,23)(19,24,27,32)(20,25,30,29)(21,26,31,28)(33,35,39,46)(34,40,44,47)(36,43,45,37)(38,42,48,41);(2,6)(4,13)(5,10)(11,16)(18,23)(19,27)(20,30)(21,31)(33,36,39,45)(34,41,44,42)(35,37,46,43)(38,40,48,47);(1,4,7,13)(2,8,6,12)(3,11,9,16)(5,15,10,14)(17,19)(18,24)(20,26)(21,29)(22,27)(23,32)(25,31)(28,30)(33,36)(34,42)(35,43)(37,46)(38,47)(39,45)(40,48)(41,44);(1,5,7,10)(2,9,6,3)(4,14,13,15)(8,11,12,16)(17,20)(18,25)(19,28)(21,24)(22,30)(23,29)(26,27)(31,32)(33,37,39,43)(34,38,44,48)(35,45,46,36)(40,41,47,42);(1,6,7,2)(3,10,9,5)(4,12,13,8)(11,14,16,15)(17,21)(18,26)(19,29)(20,24)(22,31)(23,28)(25,27)(30,32)(33,38)(34,43)(35,42)(36,47)(37,44)(39,48)(40,45)(41,46)Polycyclic presentation:
2^9|b^2=gi,c^2=gh,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8884478);