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 | |
4097 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(18,21)(23,26)(24,29)(30,32)(33,34,39,44)(35,42,37,40)(36,38,43,41)(45,48,47,46);(2,6)(7,9)(8,13)(14,15)(20,28)(24,29)(25,31)(30,32)(33,35,39,37)(34,40,44,42)(36,47,43,45)(38,48,41,46);(3,10)(4,12)(7,9)(8,13)(17,18,22,21)(19,23,27,26)(20,24,28,29)(25,30,31,32)(33,36,39,43)(34,41,44,38)(35,45,37,47)(40,48,42,46);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19,22,27)(18,23,21,26)(20,25,28,31)(24,30,29,32)(33,37,39,35)(34,42,44,40)(36,47,43,45)(38,48,41,46);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,20,22,28)(18,24,21,29)(19,25,27,31)(23,30,26,32)(33,38,39,41)(34,43,44,36)(35,46,37,48)(40,47,42,45);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,21,22,18)(19,26,27,23)(20,29,28,24)(25,32,31,30)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=h,e^2=h,f^2=hi,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, 8718373);