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 | |
4212 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)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18,22,21)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34,39,41)(35,42,38,40)(36,37,46,47)(43,44,45,48);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,32)(29,31)(33,35,39,38)(34,40,41,42)(36,45,46,43)(37,44,47,48);(2,3)(7,9)(8,10)(15,16)(20,28)(24,29)(26,32)(30,31)(34,41)(36,46)(40,42)(43,45);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20,22,28)(18,24,21,29)(19,26,27,32)(23,30,25,31)(33,36)(34,37)(35,43)(38,45)(39,46)(40,48)(41,47)(42,44);(1,4)(2,7)(3,9)(5,13)(6,12)(8,16)(10,15)(11,14)(17,19)(18,25)(20,26)(21,23)(22,27)(24,31)(28,32)(29,30)(33,37)(34,36)(35,44)(38,48)(39,47)(40,45)(41,46)(42,43);(1,5)(2,8)(3,10)(4,11)(6,14)(7,15)(9,16)(12,13)(17,21,22,18)(19,25,27,23)(20,24,28,29)(26,30,32,31)(33,38,39,35)(34,40,41,42)(36,43,46,45)(37,44,47,48)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8641524);