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 | |
4262 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)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(34,40)(35,37)(36,45)(44,48);(3,4)(5,14)(7,10)(8,13)(17,18,23,22)(19,26,20,24)(21,29,30,25)(27,31,28,32)(33,34)(35,41)(36,38)(37,47)(39,40)(42,45)(43,48)(44,46);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,19,23,20)(18,24,22,26)(21,27,30,28)(25,31,29,32)(33,35,39,37)(34,41,40,47)(36,43,45,46)(38,48,42,44);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,20,23,19)(18,24,22,26)(21,28,30,27)(25,31,29,32)(33,36)(34,42)(35,43)(37,46)(38,40)(39,45)(41,44)(47,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,21,23,30)(18,25,22,29)(19,27,20,28)(24,31,26,32)(33,37,39,35)(34,41,40,47)(36,46,45,43)(38,48,42,44);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,22,23,18)(19,24,20,26)(21,29,30,25)(27,31,28,32)(33,38,39,42)(34,36,40,45)(35,44,37,48)(41,46,47,43)Polycyclic presentation:
2^9|b^2=ghi,c^2=gi,d^2=i,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8179127);