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 | |
4131 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)(3,6)(4,7)(5,8)(9,10)(11,16)(12,15)(13,14)(17,18)(19,28)(20,24)(21,25)(22,26)(23,27)(29,32)(30,31)(33,34,38,43)(35,42,45,39)(36,46,44,40)(37,41,47,48);(1,3)(2,6)(4,10)(5,13)(7,9)(8,14)(11,15)(12,16)(17,19)(18,22)(20,29)(21,30)(23,26)(24,25)(27,28)(31,32)(33,35,38,45)(34,39,43,42)(36,37,44,47)(40,41,46,48);(2,4)(3,9)(8,12)(11,13)(17,20,23,30)(18,24,27,31)(19,29,26,21)(22,25,28,32)(33,36,38,44)(34,40,43,46)(35,37,45,47)(39,41,42,48);(1,4,7,2)(3,10,9,6)(5,8,15,12)(11,16,13,14)(17,20)(18,24)(19,29)(21,26)(22,25)(23,30)(27,31)(28,32)(33,37,38,47)(34,41,43,48)(35,44,45,36)(39,46,42,40);(1,5)(2,8)(3,11)(4,12)(6,16)(7,15)(9,13)(10,14)(17,21,23,29)(18,25,27,32)(19,30,26,20)(22,24,28,31)(33,35)(34,42)(36,37)(38,45)(39,43)(40,48)(41,46)(44,47);(1,6,7,10)(2,9,4,3)(5,14,15,16)(8,11,12,13)(17,22,23,28)(18,26,27,19)(20,25,30,32)(21,31,29,24)(33,34)(35,39)(36,40)(37,41)(38,43)(42,45)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=g,d^2=h,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, 9113820);