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 | |
4030 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)(3,7)(4,5)(6,9)(8,10)(11,12)(13,16)(14,15)(17,18,23,27)(19,21,26,24)(20,22,30,25)(28,31,32,29)(33,34,39,44)(35,45,47,40)(36,46,48,41)(37,42,43,38);(1,3)(2,7)(4,11)(5,12)(6,10)(8,9)(13,14)(15,16)(17,19,23,26)(18,24,27,21)(20,28,30,32)(22,29,25,31)(33,34,39,44)(35,45,47,40)(36,41,48,46)(37,38,43,42);(3,10)(7,8)(11,13)(12,16)(17,20,23,30)(18,25,27,22)(19,28,26,32)(21,29,24,31)(33,35,39,47)(34,40,44,45)(36,43,48,37)(38,46,42,41);(1,3)(2,8)(4,13)(5,12)(6,10)(7,9)(11,14)(15,16)(17,20)(18,22)(19,28)(21,31)(23,30)(24,29)(25,27)(26,32)(33,36,39,48)(34,41,44,46)(35,37,47,43)(38,40,42,45);(1,4,6,14)(2,5,9,15)(3,11,10,13)(7,12,8,16)(17,21)(18,26)(19,27)(20,29)(22,28)(23,24)(25,32)(30,31)(33,37)(34,42)(35,36)(38,44)(39,43)(40,41)(45,46)(47,48);(1,5)(2,4)(3,12)(6,15)(7,11)(8,13)(9,14)(10,16)(17,22)(18,20)(19,29)(21,32)(23,25)(24,28)(26,31)(27,30)(33,38,39,42)(34,43,44,37)(35,46,47,41)(36,40,48,45)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9110929);