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 | |
4124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,9)(3,7,11,6)(4,12,13,8)(10,14,16,15)(17,18,23,22)(19,24,28,27)(20,29,31,25)(21,32,30,26)(33,34,38,41)(35,39,43,36)(37,40,46,48)(42,47,45,44);(1,2,5,9)(3,7,11,6)(4,12,13,8)(10,14,16,15)(18,22)(24,27)(25,29)(26,32)(33,35)(34,39)(36,41)(37,45)(38,43)(40,44)(42,46)(47,48);(1,3,5,11)(2,6,9,7)(4,10,13,16)(8,14,12,15)(17,19,23,28)(18,24,22,27)(20,30,31,21)(25,32,29,26)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,45,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,20,23,31)(18,25,22,29)(19,21,28,30)(24,26,27,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,48,45);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,21)(18,26)(19,20)(22,32)(23,30)(24,25)(27,29)(28,31)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,46)(47,48);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,12)(11,16)(17,22,23,18)(19,27,28,24)(20,29,31,25)(21,32,30,26)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=gi,d^2=ghi,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, 8757754);