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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3843 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,22,23)(19,25,27,20)(21,29,30,24)(26,28,32,31)(33,34)(35,44)(36,38)(37,41)(39,43)(40,45)(42,48)(46,47);(1,2,6,10)(3,8,13,7)(4,5,9,16)(11,15,14,12)(18,23)(20,25)(24,29)(28,31)(35,45)(37,46)(40,44)(41,47);(1,3,6,13)(2,7,10,8)(4,14,9,11)(5,15,16,12)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)(29,30)(31,32)(33,35,39,45)(34,40,43,44)(36,37,42,46)(38,47,48,41);(1,4,6,9)(2,5,10,16)(3,11,13,14)(7,12,8,15)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32)(33,36,39,42)(34,38,43,48)(35,46,45,37)(40,41,44,47);(1,3)(2,8)(4,14)(5,12)(6,13)(7,10)(9,11)(15,16)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31)(33,37,39,46)(34,41,43,47)(35,36,45,42)(38,40,48,44);(1,5)(2,9)(3,12)(4,10)(6,16)(7,14)(8,11)(13,15)(17,21,22,30)(18,24,23,29)(19,26,27,32)(20,28,25,31)(33,38)(34,42)(35,41)(36,43)(37,40)(39,48)(44,46)(45,47)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=gi,d^2=gi,e^2=gh,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, 8901481);