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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4011 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)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18)(19,23)(20,25)(21,26)(22,27)(24,28)(29,31)(30,32)(33,34)(35,45)(36,42)(37,38)(39,44)(40,47)(41,48)(43,46);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,27)(25,31)(26,32)(33,35)(34,40)(36,37)(38,41)(39,47)(42,43)(44,45)(46,48);(2,3)(7,9)(8,10)(15,16)(19,28)(23,24)(29,32)(30,31)(33,36,39,48)(34,41,44,42)(35,37,47,46)(38,45,43,40);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,19)(18,24)(20,29)(21,31)(22,28)(23,27)(25,30)(26,32)(33,36)(34,42)(35,46)(37,47)(38,40)(39,48)(41,44)(43,45);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,20)(18,25)(19,29)(21,27)(22,26)(23,31)(24,30)(28,32)(33,37,39,46)(34,43,44,38)(35,36,47,48)(40,42,45,41);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21,22,25)(18,26,27,20)(19,30,28,31)(23,32,24,29)(33,38,39,43)(34,37,44,46)(35,41,47,42)(36,45,48,40)Polycyclic presentation:
2^9|c^2=g,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, 8653778);