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 | |
4107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,7,10,14)(4,5,8,13)(11,12,15,16)(17,18,23,25)(19,24,28,20)(21,30,29,26)(22,27,31,32)(33,34)(35,42)(36,39)(37,44)(38,40)(41,46)(43,47)(45,48);(1,3,6,10)(2,7,9,14)(4,11,8,15)(5,12,13,16)(17,19)(18,24)(20,25)(21,31)(22,29)(23,28)(26,27)(30,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(3,10)(7,14)(11,15)(12,16)(18,25)(19,28)(21,29)(27,32)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(1,4)(2,5)(3,11)(6,8)(7,12)(9,13)(10,15)(14,16)(17,20)(18,19)(21,27)(22,26)(23,24)(25,28)(29,32)(30,31)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,3)(2,7)(4,11)(5,12)(6,10)(8,15)(9,14)(13,16)(17,21,23,29)(18,26,25,30)(19,22,28,31)(20,27,24,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5)(2,8)(3,12)(4,9)(6,13)(7,15)(10,16)(11,14)(17,22)(18,27)(19,29)(20,30)(21,28)(23,31)(24,26)(25,32)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=g,d^2=g,e^2=gh,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, 8503162);