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 |
| prime radical | #54942 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,10,8)(4,12,13,9)(11,16,14,15)(19,25)(23,26)(27,31)(30,32)(33,34)(35,37)(36,44)(38,39)(40,43)(41,47)(42,48)(45,46);(2,7)(4,13)(5,8)(11,14)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(34,39)(37,40)(41,44)(46,48);(2,7)(3,10)(9,12)(11,14)(18,22)(23,26)(24,28)(30,32)(33,35,38,43)(34,37,39,40)(36,42,47,45)(41,46,44,48);(1,3)(2,8)(4,14)(5,7)(6,10)(9,15)(11,13)(12,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35)(34,40)(36,45)(37,39)(38,43)(41,46)(42,47)(44,48);(1,4,6,13)(2,9,7,12)(3,11,10,14)(5,15,8,16)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,36,38,47)(34,41,39,44)(35,42,43,45)(37,46,40,48);(1,5,6,8)(2,3,7,10)(4,15,13,16)(9,11,12,14)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,37,38,40)(34,35,39,43)(36,46,47,48)(41,42,44,45)Polycyclic presentation:
2^9|a^2=h,c^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8761537);