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 | |
4228 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)(17,18,23,21)(19,26,25,22)(20,28,29,24)(27,30,31,32)(33,34,39,40)(35,41,36,38)(37,45,47,42)(43,46,44,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,32)(29,31)(34,40)(35,36)(42,45)(43,44);(2,6)(7,9)(8,13)(14,15)(20,29)(24,28)(27,31)(30,32)(33,35,39,36)(34,38,40,41)(37,43,47,44)(42,46,45,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20)(18,24)(19,27)(21,28)(22,30)(23,29)(25,31)(26,32)(33,36,39,35)(34,41,40,38)(37,44,47,43)(42,48,45,46);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,21,23,18)(19,26,25,22)(20,24,29,28)(27,30,31,32)(33,37,39,47)(34,42,40,45)(35,43,36,44)(38,46,41,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,22,23,26)(18,25,21,19)(20,30,29,32)(24,31,28,27)(33,38,39,41)(34,35,40,36)(37,46,47,48)(42,43,45,44)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,e^2=g,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, 8062242);