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 | |
4206 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,22,23)(19,20,24,27)(21,25,30,31)(26,28,32,29)(33,34,39,41)(35,45,46,40)(36,37,43,42)(38,48,47,44);(3,9)(7,13)(10,15)(14,16)(18,23)(20,27)(25,31)(28,29)(33,35)(34,40)(36,47)(37,48)(38,43)(39,46)(41,45)(42,44);(2,6)(7,13)(8,11)(14,16)(34,41)(35,46)(36,43)(44,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19,22,24)(18,20,23,27)(21,26,30,32)(25,28,31,29)(33,36,39,43)(34,42,41,37)(35,47,46,38)(40,44,45,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20)(18,24)(19,23)(21,29)(22,27)(25,26)(28,30)(31,32)(33,37,39,42)(34,43,41,36)(35,44,46,48)(38,40,47,45);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,30)(18,25,23,31)(19,26,24,32)(20,28,27,29)(33,38,39,47)(34,44,41,48)(35,43,46,36)(37,40,42,45)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9817936);