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 | |
4009 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)(3,10)(4,7)(5,13)(6,9)(8,15)(11,12)(14,16)(19,27)(20,29)(22,28)(24,32)(33,34,39,40)(35,43,44,41)(36,47,48,37)(38,45,46,42);(1,3,6,7)(2,4,9,10)(5,11,15,14)(8,12,13,16)(17,18,23,26)(19,28,27,22)(20,24,29,32)(21,25,31,30)(34,40)(35,44)(37,47)(42,45);(1,3,6,7)(2,4,9,10)(5,14,15,11)(8,16,13,12)(17,19,23,27)(18,22,26,28)(20,21,29,31)(24,30,32,25)(33,35,39,44)(34,41,40,43)(36,45,48,42)(37,46,47,38);(1,2)(3,4)(5,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,20)(18,24)(19,21)(22,30)(23,29)(25,28)(26,32)(27,31)(33,36)(34,37)(35,45)(38,43)(39,48)(40,47)(41,46)(42,44);(1,4)(2,7)(3,9)(5,12)(6,10)(8,14)(11,13)(15,16)(17,21)(18,25)(19,29)(20,27)(22,24)(23,31)(26,30)(28,32)(33,37)(34,36)(35,46)(38,44)(39,47)(40,48)(41,45)(42,43);(1,5,6,15)(2,8,9,13)(3,11,7,14)(4,12,10,16)(17,22)(18,19)(20,30)(21,24)(23,28)(25,29)(26,27)(31,32)(33,38,39,46)(34,42,40,45)(35,37,44,47)(36,43,48,41)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8833734);