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 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(7,13)(10,16)(14,15)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,34,39,44)(35,36,41,40)(37,42,46,48)(38,43,47,45);(1,2)(3,8)(4,13)(5,9)(6,11)(7,12)(10,14)(15,16)(17,18,23,20)(19,24,22,26)(21,25,30,28)(27,31,29,32)(33,35,39,41)(34,40,44,36)(37,45,46,43)(38,42,47,48);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,16)(13,15)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,36,39,40)(34,41,44,35)(37,38,46,47)(42,43,48,45);(1,4)(2,7)(3,10)(5,12)(6,14)(8,15)(9,13)(11,16)(17,20,23,18)(19,26,22,24)(21,25,30,28)(27,31,29,32)(33,35)(34,36)(37,45)(38,42)(39,41)(40,44)(43,46)(47,48);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,16)(13,14)(17,21,23,30)(18,25,20,28)(19,27,22,29)(24,31,26,32)(33,37)(34,42)(35,45)(36,38)(39,46)(40,47)(41,43)(44,48);(1,2)(3,6)(4,7)(5,9)(8,11)(10,14)(12,13)(15,16)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,38)(34,43)(35,42)(36,46)(37,40)(39,47)(41,48)(44,45)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10362861);