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 | |
4097 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)(3,8)(4,14)(5,15)(6,9)(7,13)(10,16)(11,12)(17,18)(19,20)(21,29)(22,23)(24,27)(25,31)(26,32)(28,30)(33,34,40,37)(35,44,46,41)(36,42,47,38)(39,43,45,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,9)(10,15)(14,16)(18,23)(20,24)(25,29)(28,32)(33,35)(34,41)(36,43)(37,44)(38,45)(39,42)(40,46)(47,48);(1,4)(2,5)(3,10)(6,15)(7,16)(8,12)(9,14)(11,13)(17,19)(18,24)(20,23)(21,26)(22,27)(25,32)(28,29)(30,31)(33,36,40,47)(34,42,37,38)(35,43,46,48)(39,44,45,41);(1,4)(2,5)(3,11)(6,15)(7,12)(8,16)(9,14)(10,13)(17,20,22,24)(18,19,23,27)(21,28,31,32)(25,26,29,30)(33,37,40,34)(35,41,46,44)(36,38,47,42)(39,48,45,43);(1,3)(2,8)(4,10)(5,12)(6,13)(7,9)(11,15)(14,16)(17,19)(18,20)(21,30)(22,27)(23,24)(25,32)(26,31)(28,29)(33,38)(34,36)(35,45)(37,47)(39,46)(40,42)(41,43)(44,48);(1,5)(2,4)(3,12)(6,14)(7,11)(8,10)(9,15)(13,16)(17,21)(18,25)(19,26)(20,28)(22,31)(23,29)(24,32)(27,30)(33,39,40,45)(34,43,37,48)(35,42,46,38)(36,44,47,41)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10389673);