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 | |
4154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,9)(4,13)(5,14)(6,8)(7,12)(10,16)(11,15)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,28)(29,32)(33,34)(35,36)(37,40)(38,46)(39,42)(41,48)(43,45)(44,47);(1,3,6,12)(2,7,8,9)(4,10,14,15)(5,16,13,11)(17,19)(18,23)(20,21)(22,27)(24,30)(25,32)(26,28)(29,31)(33,35,39,40)(34,36,42,37)(38,47,48,43)(41,45,46,44);(2,8)(4,14)(7,9)(10,15)(19,27)(20,29)(23,28)(24,32)(33,36,39,37)(34,40,42,35)(38,44,48,45)(41,47,46,43);(1,4)(2,5)(3,10)(6,14)(7,16)(8,13)(9,11)(12,15)(17,20,22,29)(18,24,26,32)(19,21,27,31)(23,30,28,25)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,3)(2,7)(4,15)(5,11)(6,12)(8,9)(10,14)(13,16)(17,21)(18,25)(19,20)(22,31)(23,32)(24,28)(26,30)(27,29)(33,37,39,36)(34,35,42,40)(38,45,48,44)(41,43,46,47);(1,5)(2,4)(3,11)(6,13)(7,15)(8,14)(9,10)(12,16)(17,18)(19,28)(20,24)(21,25)(22,26)(23,27)(29,32)(30,31)(33,38,39,48)(34,41,42,46)(35,43,40,47)(36,44,37,45)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8746403);