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 | |
4057 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,10,5)(4,12,13,8)(11,15,14,16)(18,19)(23,25)(24,26)(30,31)(33,34,39,43)(35,40,44,48)(36,37,41,47)(38,42,46,45);(2,7)(3,10)(4,13)(15,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,35)(34,40)(36,45)(37,38)(39,44)(41,42)(43,48)(46,47);(2,7)(5,9)(8,12)(15,16)(21,29)(24,26)(27,32)(30,31)(35,44)(38,46)(40,48)(42,45);(1,3)(2,5)(4,14)(6,10)(7,9)(8,16)(11,13)(12,15)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31)(33,36)(34,37)(35,45)(38,40)(39,41)(42,44)(43,47)(46,48);(1,4)(2,8)(3,11)(5,15)(6,13)(7,12)(9,16)(10,14)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,37)(34,41)(35,46)(36,43)(38,44)(39,47)(40,45)(42,48);(1,5)(2,3)(4,15)(6,9)(7,10)(8,11)(12,14)(13,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,38,39,46)(34,42,43,45)(35,37,44,47)(36,40,41,48)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,d^2=gh,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8281413);