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 | |
4067 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18,22,26)(19,20,24,23)(21,25,30,32)(27,28,29,31)(33,34,39,40)(35,43,44,41)(36,46,45,42)(37,38,47,48);(2,6)(4,13)(7,9)(11,16)(17,19,22,24)(18,23,26,20)(21,29,30,27)(25,28,32,31)(34,40)(35,44)(37,47)(42,46);(2,6)(3,10)(8,12)(11,16)(17,18,22,26)(19,23,24,20)(21,25,30,32)(27,31,29,28)(33,35,39,44)(34,41,40,43)(36,37,45,47)(38,46,48,42);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,29)(30,32)(33,36,39,45)(34,42,40,46)(35,37,44,47)(38,43,48,41);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,23)(18,24,26,19)(21,28,30,31)(25,29,32,27)(33,37)(34,38)(35,45)(36,44)(39,47)(40,48)(41,46)(42,43);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21,22,30)(18,25,26,32)(19,27,24,29)(20,28,23,31)(33,38)(34,37)(35,46)(36,43)(39,48)(40,47)(41,45)(42,44)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8680236);