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 | |
4028 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)(3,8)(4,12)(5,7)(6,11)(9,14)(10,15)(13,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,29)(30,31)(33,34)(35,39)(36,37)(38,41)(40,44)(42,43)(45,48)(46,47);(1,3,5,11)(2,6,7,8)(4,10,14,13)(9,15,12,16)(17,19,22,24)(18,21,25,26)(20,27,28,31)(23,30,29,32)(33,35)(34,39)(36,42)(37,43)(38,44)(40,41)(45,46)(47,48);(2,7)(6,8)(9,12)(15,16)(20,28)(23,29)(27,31)(30,32)(33,35,38,44)(34,39,41,40)(36,42,46,45)(37,43,47,48);(1,2)(3,6)(4,9)(5,7)(8,11)(10,15)(12,14)(13,16)(17,20,22,28)(18,23,25,29)(19,27,24,31)(21,30,26,32)(33,35)(34,40)(36,45)(37,43)(38,44)(39,41)(42,46)(47,48);(1,3)(2,8)(4,13)(5,11)(6,7)(9,15)(10,14)(12,16)(17,18)(19,21)(20,29)(22,25)(23,28)(24,26)(27,32)(30,31)(33,36,38,46)(34,37,41,47)(35,42,44,45)(39,43,40,48);(1,4)(2,9)(3,10)(5,14)(6,15)(7,12)(8,16)(11,13)(17,21)(18,24)(19,25)(20,30)(22,26)(23,31)(27,29)(28,32)(33,37)(34,36)(35,43)(38,47)(39,42)(40,45)(41,46)(44,48)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=gh,e^2=i,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, 9113942);