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 | |
4159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(10,16)(14,15)(17,18,22,26)(19,27,29,23)(20,24,30,28)(21,31,32,25)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19)(18,23)(20,25)(21,28)(22,29)(24,32)(26,27)(30,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(2,6)(7,9)(8,13)(14,15)(17,18,22,26)(19,23,29,27)(20,24,30,28)(21,25,32,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18)(19,23)(20,28)(21,25)(22,26)(24,30)(27,29)(31,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20)(18,24)(19,25)(21,27)(22,30)(23,32)(26,28)(29,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,32)(18,25,26,31)(19,28,29,24)(20,27,30,23)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8062230);