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 | |
4285 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,11)(3,8,12,7)(4,9,15,14)(5,10,16,13)(20,28)(24,29)(27,32)(30,31)(33,34)(35,42)(36,41)(37,46)(38,39)(40,45)(43,44)(47,48);(3,12)(4,15)(7,8)(9,14)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,30)(31,32)(33,35)(34,40)(36,43)(37,44)(38,42)(39,45)(41,47)(46,48);(1,3)(2,7)(4,13)(5,14)(6,12)(8,11)(9,16)(10,15)(17,19)(18,21)(20,27)(22,25)(23,26)(24,30)(28,32)(29,31)(33,36,39,46)(34,41,38,37)(35,43,45,48)(40,47,42,44);(1,3)(2,8)(4,13)(5,9)(6,12)(7,11)(10,15)(14,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,32)(29,30)(33,37)(34,36)(35,44)(38,46)(39,41)(40,43)(42,48)(45,47);(1,4)(2,9)(3,13)(5,8)(6,15)(7,16)(10,12)(11,14)(17,20,22,28)(18,24,26,29)(19,27,25,32)(21,30,23,31)(33,38,39,34)(35,40,45,42)(36,37,46,41)(43,47,48,44);(1,5)(2,10)(3,14)(4,8)(6,16)(7,15)(9,12)(11,13)(17,21,22,23)(18,25,26,19)(20,30,28,31)(24,32,29,27)(33,35)(34,42)(36,43)(37,47)(38,40)(39,45)(41,44)(46,48)Polycyclic presentation:
2^9|a^2=h,c^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8413491);