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 | |
4266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,11,12,7)(4,13,14,9)(5,10,15,16)(17,18,22,25)(19,26,21,23)(20,24,28,31)(27,32,29,30)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48);(1,3,6,12)(2,7,8,11)(4,10,14,16)(5,9,15,13)(17,19,22,21)(18,23,25,26)(20,27,28,29)(24,30,31,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,47)(45,48);(2,8)(4,14)(7,11)(10,16)(20,28)(24,31)(27,29)(30,32)(33,35,37,43)(34,39,41,38)(36,42,45,48)(40,47,44,46);(1,3)(2,7)(4,10)(5,13)(6,12)(8,11)(9,15)(14,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,34)(35,39)(36,44)(37,41)(38,43)(40,45)(42,46)(47,48);(1,4,6,14)(2,9,8,13)(3,10,12,16)(5,7,15,11)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,5,6,15)(2,10,8,16)(3,9,12,13)(4,7,14,11)(17,21,22,19)(18,23,25,26)(20,29,28,27)(24,30,31,32)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,44)(43,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8441912);