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 | |
4107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
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)(17,18,23,25)(19,27,21,24)(20,26,30,31)(22,28,32,29)(33,34,39,43)(35,40,38,42)(36,41,47,37)(44,46,45,48);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19,23,21)(18,24,25,27)(20,29,30,28)(22,26,32,31)(33,35,39,38)(34,40,43,42)(36,45,47,44)(37,46,41,48);(2,3)(7,9)(8,10)(15,16)(18,25)(20,30)(24,27)(28,29)(33,36,39,47)(34,41,43,37)(35,44,38,45)(40,46,42,48);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20)(18,26)(19,28)(21,29)(22,24)(23,30)(25,31)(27,32)(33,36)(34,37)(35,45)(38,44)(39,47)(40,46)(41,43)(42,48);(1,4)(2,7)(3,9)(5,13)(6,12)(8,16)(10,15)(11,14)(17,21,23,19)(18,27,25,24)(20,29,30,28)(22,26,32,31)(33,37)(34,36)(35,46)(38,48)(39,41)(40,45)(42,44)(43,47);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,22,23,32)(18,28,25,29)(19,26,21,31)(20,27,30,24)(33,38,39,35)(34,42,43,40)(36,45,47,44)(37,46,41,48)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=ghi,d^2=gh,e^2=g,f^2=gi,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, 8657247);