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 | |
4022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,23,24)(19,21,20,25)(22,26,30,31)(27,29,28,32)(33,34)(35,36)(37,47)(38,41)(39,42)(40,46)(43,48)(44,45);(1,2,6,7)(3,5,9,14)(4,8,12,15)(10,13,16,11)(18,24)(19,20)(26,31)(27,28)(33,35,39,40)(34,36,42,46)(37,43,41,45)(38,44,47,48);(2,7)(3,9)(8,15)(10,16)(17,19,23,20)(18,25,24,21)(22,27,30,28)(26,32,31,29)(33,36,39,46)(34,40,42,35)(37,44,41,48)(38,45,47,43);(1,3)(2,5)(4,10)(6,9)(7,14)(8,13)(11,15)(12,16)(17,20,23,19)(18,21,24,25)(22,28,30,27)(26,29,31,32)(33,36)(34,35)(37,44)(38,43)(39,46)(40,42)(41,48)(45,47);(1,4)(2,8)(3,10)(5,13)(6,12)(7,15)(9,16)(11,14)(17,21)(18,19)(20,24)(22,29)(23,25)(26,27)(28,31)(30,32)(33,37)(34,38)(35,43)(36,44)(39,41)(40,45)(42,47)(46,48);(1,5)(2,9)(3,7)(4,11)(6,14)(8,10)(12,13)(15,16)(17,22,23,30)(18,26,24,31)(19,27,20,28)(21,29,25,32)(33,38)(34,41)(35,44)(36,45)(37,42)(39,47)(40,48)(43,46)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=h,d^2=g,f^2=g,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, 8616333);