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 | |
4011 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,5,10,9)(13,14,16,15)(33,34)(35,37)(36,46)(38,41)(39,42)(40,44)(43,45)(47,48);(2,7)(3,12)(4,10)(14,15)(17,18,22,23)(19,24,28,30)(20,27,29,25)(21,26,31,32)(33,35,39,40)(34,37,42,44)(36,47,41,43)(38,45,46,48);(2,7)(4,10)(8,11)(13,16)(18,23)(20,29)(24,30)(26,32)(33,36,39,41)(34,38,42,46)(35,43,40,47)(37,48,44,45);(1,3)(2,8)(4,13)(5,15)(6,12)(7,11)(9,14)(10,16)(17,19)(18,24)(20,26)(21,25)(22,28)(23,30)(27,31)(29,32)(33,34)(35,44)(36,38)(37,40)(39,42)(41,46)(43,45)(47,48);(1,4,6,10)(2,9,7,5)(3,13,12,16)(8,14,11,15)(17,20,22,29)(18,25,23,27)(19,26,28,32)(21,30,31,24)(33,37,39,44)(34,40,42,35)(36,45,41,48)(38,43,46,47);(1,5,6,9)(2,10,7,4)(3,14,12,15)(8,13,11,16)(17,21)(18,26)(19,27)(20,30)(22,31)(23,32)(24,29)(25,28)(33,38)(34,41)(35,45)(36,42)(37,43)(39,46)(40,48)(44,47)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,e^2=gi,f^2=g,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, 8525214);