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 | |
4145 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(7,14)(8,11)(15,16)(17,18,22,23)(19,25,20,24)(21,26,30,29)(27,32,28,31)(33,34)(35,39)(36,45)(37,40)(38,42)(41,44)(43,48)(46,47);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(18,23)(24,25)(26,29)(31,32)(33,35)(34,39)(36,46)(37,43)(38,44)(40,48)(41,42)(45,47);(17,19,22,20)(18,24,23,25)(21,27,30,28)(26,31,29,32)(33,36,38,37)(34,40,42,45)(35,43,44,46)(39,47,41,48);(1,2)(3,7)(4,11)(5,6)(8,13)(9,16)(10,14)(12,15)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,32)(28,31)(33,34)(35,41)(36,40)(37,45)(38,42)(39,44)(43,48)(46,47);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,20,22,19)(18,25,23,24)(21,27,30,28)(26,31,29,32)(33,35)(34,41)(36,43)(37,46)(38,44)(39,42)(40,48)(45,47);(1,4)(2,8)(3,9)(5,13)(6,11)(7,15)(10,12)(14,16)(17,21,22,30)(18,26,23,29)(19,27,20,28)(24,31,25,32)(33,37,38,36)(34,40,42,45)(35,43,44,46)(39,48,41,47)Polycyclic presentation:
2^9|a^2=hi,c^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8570149);