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 | |
4272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(18,23)(19,26)(25,30)(27,32)(33,34,39,37)(35,40,38,45)(36,41,43,46)(42,47,44,48);(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,18,22,23)(19,24,26,20)(21,25,29,30)(27,31,32,28)(33,35,39,38)(34,40,37,45)(36,42,43,44)(41,47,46,48);(2,3)(4,11)(5,13)(14,15)(19,26)(20,24)(21,29)(25,30)(34,37)(35,38)(36,43)(47,48);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19,22,26)(18,24,23,20)(21,27,29,32)(25,31,30,28)(33,36)(34,41)(35,42)(37,46)(38,44)(39,43)(40,47)(45,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20,22,24)(18,19,23,26)(21,28,29,31)(25,27,30,32)(33,37,39,34)(35,40,38,45)(36,41,43,46)(42,48,44,47);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,29)(18,25,23,30)(19,27,26,32)(20,28,24,31)(33,38,39,35)(34,40,37,45)(36,44,43,42)(41,47,46,48)Polycyclic presentation:
2^9|a^2=h,b^2=gi,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8897510);