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 | |
4187 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)(3,11)(4,9)(5,15)(6,7)(8,12)(10,16)(13,14)(33,34)(35,39)(36,40)(37,41)(38,43)(42,47)(44,46)(45,48);(2,7)(3,12)(9,13)(10,15)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,31)(30,32)(33,35,37,43)(34,38,41,39)(36,44,45,42)(40,47,48,46);(2,7)(3,12)(4,14)(5,16)(18,22)(19,26)(20,29)(31,32)(33,35,37,43)(34,39,41,38)(36,44,45,42)(40,46,48,47);(1,3)(2,8)(4,15)(5,13)(6,12)(7,11)(9,16)(10,14)(17,19)(18,23)(20,30)(21,26)(22,25)(24,32)(27,29)(28,31)(33,35)(34,38)(36,44)(37,43)(39,41)(40,47)(42,45)(46,48);(1,4,6,14)(2,9,7,13)(3,10,12,15)(5,11,16,8)(17,20)(18,24)(19,27)(21,29)(22,28)(23,31)(25,32)(26,30)(33,36,37,45)(34,40,41,48)(35,42,43,44)(38,46,39,47);(1,5)(2,10)(3,13)(4,11)(6,16)(7,15)(8,14)(9,12)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,47)(45,48)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8965468);