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 | |
4177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(7,9)(8,12)(14,15)(17,18)(19,23)(20,22)(21,24)(25,27)(26,31)(28,30)(29,32)(33,34,38,40)(35,42,37,39)(36,44,46,41)(43,47,45,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,18)(19,25)(20,22)(21,24)(23,27)(26,32)(28,30)(29,31)(33,35,38,37)(34,39,40,42)(36,45,46,43)(41,48,44,47);(3,10)(7,9)(11,16)(14,15)(18,20)(23,25)(24,28)(31,32)(34,40)(39,42)(41,44)(47,48);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,36)(34,41)(35,43)(37,45)(38,46)(39,47)(40,44)(42,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,20,22,18)(19,25,27,23)(21,28,30,24)(26,32,29,31)(33,34)(35,42)(36,41)(37,39)(38,40)(43,48)(44,46)(45,47);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,37,38,35)(34,39,40,42)(36,43,46,45)(41,48,44,47)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8529211);