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 | |
4112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,11)(7,9)(12,16)(14,15)(17,18)(19,21)(20,28)(22,23)(24,27)(25,31)(26,32)(29,30)(33,34,39,44)(35,42,37,40)(36,41,47,45)(38,46,48,43);(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(18,23)(21,24)(25,28)(30,32)(35,37)(38,48)(40,42)(43,46);(2,3)(7,9)(8,10)(14,15)(17,19,22,27)(18,21,23,24)(20,26,31,29)(25,30,28,32)(33,35,39,37)(34,40,44,42)(36,46,47,43)(38,45,48,41);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19)(18,24)(20,29)(21,23)(22,27)(25,30)(26,31)(28,32)(33,36)(34,41)(35,43)(37,46)(38,42)(39,47)(40,48)(44,45);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20)(18,25)(19,26)(21,30)(22,31)(23,28)(24,32)(27,29)(33,37,39,35)(34,42,44,40)(36,43,47,46)(38,41,48,45);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,24)(18,19,23,27)(20,30,31,32)(25,26,28,29)(33,38,39,48)(34,43,44,46)(35,45,37,41)(36,42,47,40)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=gh,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8750783);