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 | |
4281 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,10,8)(4,5,13,15)(11,12,14,16)(33,34)(35,42)(36,39)(37,40)(38,41)(43,45)(44,47)(46,48);(2,7)(4,13)(8,9)(11,14)(17,18,22,23)(19,24,26,31)(20,21,25,30)(27,32,29,28)(33,35)(34,36)(37,43)(38,39)(40,44)(41,42)(45,48)(46,47);(2,7)(3,10)(5,15)(11,14)(18,23)(19,26)(21,30)(27,29)(33,36,38,42)(34,39,41,35)(37,44,46,45)(40,47,48,43);(1,3)(2,8)(4,14)(5,12)(6,10)(7,9)(11,13)(15,16)(17,19)(18,24)(20,29)(21,28)(22,26)(23,31)(25,27)(30,32)(33,36)(34,35)(37,45)(38,42)(39,41)(40,47)(43,48)(44,46);(1,4)(2,5)(3,11)(6,13)(7,15)(8,16)(9,12)(10,14)(17,20,22,25)(18,21,23,30)(19,27,26,29)(24,32,31,28)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,48)(42,45);(1,5)(2,4)(3,12)(6,15)(7,13)(8,14)(9,11)(10,16)(17,21)(18,25)(19,28)(20,23)(22,30)(24,27)(26,32)(29,31)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8825051);