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 | |
4077 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,11,9)(4,12,13,8)(10,15,16,14)(19,25)(23,26)(27,31)(30,32)(33,34)(35,38)(36,45)(37,41)(39,43)(40,46)(42,47)(44,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30)(33,35,39,42)(34,38,43,47)(36,37,46,44)(40,48,45,41);(2,6)(7,9)(8,12)(14,15)(18,20)(23,26)(24,28)(30,32)(33,36,39,46)(34,40,43,45)(35,44,42,37)(38,41,47,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,36)(34,40)(35,44)(37,42)(38,41)(39,46)(43,45)(47,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,15)(13,16)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32)(33,37,39,44)(34,41,43,48)(35,36,42,46)(38,45,47,40);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,38)(34,42)(35,43)(36,41)(37,40)(39,47)(44,45)(46,48)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gh,d^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9976312);