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 | |
4126 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)(3,4)(5,13)(6,7)(8,15)(9,12)(10,16)(11,14)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28)(33,34,39,44)(35,45,46,40)(36,37,42,41)(38,43,47,48);(1,3,6,9)(2,4,7,12)(5,14,15,10)(8,16,13,11)(19,27)(21,31)(23,32)(25,28)(33,35)(34,40)(36,47)(37,43)(38,42)(39,46)(41,48)(44,45);(2,7)(3,9)(8,13)(10,14)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,34,39,44)(35,40,46,45)(36,41,42,37)(38,43,47,48);(1,4)(2,3)(5,11)(6,12)(7,9)(8,10)(13,14)(15,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,36)(34,41)(35,38)(37,44)(39,42)(40,43)(45,48)(46,47);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,15)(14,16)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32)(33,37,39,41)(34,42,44,36)(35,43,46,48)(38,45,47,40);(1,5)(2,8)(3,10)(4,11)(6,15)(7,13)(9,14)(12,16)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)(33,38)(34,43)(35,36)(37,45)(39,47)(40,41)(42,46)(44,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10231814);