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 | |
4115 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,11,16)(4,6,14,15)(5,13,12,10)(17,18)(19,28)(20,21)(22,29)(23,27)(24,31)(25,32)(26,30)(33,34,38,42)(35,41,37,39)(36,44,46,40)(43,47,45,48);(1,3)(2,8)(4,5)(6,13)(7,11)(9,16)(10,15)(12,14)(17,19)(18,24)(20,29)(21,30)(22,25)(23,31)(26,32)(27,28)(33,34)(35,39)(36,44)(37,41)(38,42)(40,46)(43,48)(45,47);(2,9)(3,11)(4,14)(10,13)(17,20,23,32)(18,21,27,25)(19,29,31,26)(22,24,30,28)(33,35,38,37)(34,39,42,41)(36,45,46,43)(40,48,44,47);(1,4)(2,6)(3,12)(5,11)(7,14)(8,10)(9,15)(13,16)(17,20)(18,25)(19,29)(21,27)(22,24)(23,32)(26,31)(28,30)(33,36,38,46)(34,40,42,44)(35,43,37,45)(39,47,41,48);(1,5,7,12)(2,10,9,13)(3,4,11,14)(6,8,15,16)(17,21,23,25)(18,20,27,32)(19,30,31,22)(24,29,28,26)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,46)(45,48);(1,6)(2,4)(3,13)(5,8)(7,15)(9,14)(10,11)(12,16)(17,22,23,30)(18,26,27,29)(19,25,31,21)(20,24,32,28)(33,37,38,35)(34,41,42,39)(36,45,46,43)(40,48,44,47)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014434);