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 | |
4223 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,4,7,10)(5,8,14,16)(11,12,15,13)(17,18)(19,22)(20,27)(21,24)(23,28)(25,31)(26,30)(29,32)(33,34)(35,44)(36,46)(37,41)(38,45)(39,40)(42,47)(43,48);(1,3)(2,4)(5,11)(6,7)(8,12)(9,10)(13,16)(14,15)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31)(34,40)(35,37)(42,46)(43,45);(3,7)(4,10)(11,15)(12,13)(17,18,21,24)(19,22,26,30)(20,23,28,27)(25,29,32,31)(33,35,39,37)(34,41,40,44)(36,43,47,45)(38,42,48,46);(1,2)(3,10)(4,7)(5,8)(6,9)(11,13)(12,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,36,39,47)(34,42,40,46)(35,45,37,43)(38,44,48,41);(1,4,6,10)(2,7,9,3)(5,13,14,12)(8,11,16,15)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,37,39,35)(34,41,40,44)(36,43,47,45)(38,46,48,42);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,16)(10,13)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31)(33,38)(34,43)(35,42)(36,44)(37,46)(39,48)(40,45)(41,47)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=h,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9962882);