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 | |
4198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,11,9)(4,12,13,8)(10,15,16,14)(17,18)(19,27)(20,29)(21,26)(22,30)(23,25)(24,28)(31,32)(33,34,40,43)(35,44,47,41)(36,38,37,48)(39,42,45,46);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,23,28)(18,24,25,27)(20,26,30,32)(21,29,31,22)(33,35)(34,41)(36,39)(37,45)(38,46)(40,47)(42,48)(43,44);(2,6)(7,9)(8,12)(14,15)(18,25)(20,30)(24,27)(26,32)(33,36,40,37)(34,38,43,48)(35,39,47,45)(41,46,44,42);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20)(18,22)(19,26)(21,24)(23,30)(25,29)(27,31)(28,32)(33,37,40,36)(34,38,43,48)(35,45,47,39)(41,46,44,42);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,21,23,31)(18,26,25,32)(19,22,28,29)(20,27,30,24)(33,38)(34,37)(35,46)(36,43)(39,44)(40,48)(41,45)(42,47);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,22)(18,20)(19,21)(23,29)(24,26)(25,30)(27,32)(28,31)(33,39,40,45)(34,42,43,46)(35,36,47,37)(38,41,48,44)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10009443);