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 | |
4122 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)(3,8)(4,12)(5,9)(6,11)(7,13)(10,14)(15,16)(17,18,22,23)(19,24,28,27)(20,30,31,25)(21,26,32,29)(34,39)(35,44)(36,43)(46,47);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,18,22,23)(19,27,28,24)(20,25,31,30)(21,26,32,29)(33,34)(35,40)(36,42)(37,43)(38,39)(41,44)(45,47)(46,48);(4,13)(7,12)(10,16)(14,15)(18,23)(19,28)(25,30)(26,29)(33,35,38,44)(34,40,39,41)(36,45,43,48)(37,46,42,47);(1,4)(2,7)(3,10)(5,13)(6,14)(8,15)(9,12)(11,16)(17,19,22,28)(18,24,23,27)(20,29,31,26)(21,25,32,30)(33,35)(34,41)(36,45)(37,47)(38,44)(39,40)(42,46)(43,48);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,14)(13,16)(17,20)(18,25)(19,29)(21,27)(22,31)(23,30)(24,32)(26,28)(33,36)(34,42)(35,45)(37,39)(38,43)(40,47)(41,46)(44,48);(1,2)(3,6)(4,12)(5,9)(7,13)(8,11)(10,15)(14,16)(17,21,22,32)(18,26,23,29)(19,30,28,25)(20,24,31,27)(33,37,38,42)(34,43,39,36)(35,46,44,47)(40,48,41,45)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627916);