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 | |
4201 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,14)(4,8,13,16)(5,9,15,12)(17,18)(19,27)(20,23)(21,25)(22,30)(24,28)(26,32)(29,31)(33,34)(35,38)(36,41)(37,43)(39,44)(40,47)(42,46)(45,48);(1,3)(2,7)(4,12)(5,8)(6,11)(9,13)(10,14)(15,16)(17,19,23,28)(18,24,20,27)(21,30,31,26)(22,25,32,29)(33,35)(34,40)(36,45)(37,46)(38,44)(39,47)(41,43)(42,48);(3,11)(4,13)(7,14)(8,16)(18,20)(22,32)(24,27)(25,29)(33,36,39,46)(34,41,44,42)(35,45,47,37)(38,48,40,43);(1,4)(2,8)(3,12)(5,7)(6,13)(9,11)(10,16)(14,15)(17,20,23,18)(19,27,28,24)(21,25,31,29)(22,30,32,26)(33,36)(34,42)(35,45)(37,47)(38,43)(39,46)(40,48)(41,44);(1,5)(2,9)(3,8)(4,14)(6,15)(7,13)(10,12)(11,16)(17,21,23,31)(18,25,20,29)(19,26,28,30)(22,27,32,24)(33,37,39,45)(34,43,44,48)(35,46,47,36)(38,42,40,41);(1,3)(2,7)(4,12)(5,8)(6,11)(9,13)(10,14)(15,16)(17,22,23,32)(18,26,20,30)(19,25,28,29)(21,27,31,24)(33,38,39,40)(34,35,44,47)(36,43,46,48)(37,42,45,41)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gh,d^2=h,e^2=g,f^2=g,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, 8812472);