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 | |
4172 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)(3,9)(4,12)(5,6)(7,11)(8,14)(10,15)(13,16)(18,22)(23,26)(24,28)(30,32)(33,34,39,43)(35,40,44,46)(36,38,42,48)(37,41,47,45);(2,6)(7,9)(8,12)(15,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,35)(34,40)(36,45)(37,38)(39,44)(41,42)(43,46)(47,48);(2,6)(7,9)(8,12)(15,16)(19,25)(23,26)(27,31)(30,32)(35,44)(37,47)(40,46)(41,45);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,18)(19,26)(20,28)(21,22)(23,25)(24,29)(27,30)(31,32)(33,36)(34,38)(35,45)(37,40)(39,42)(41,44)(43,48)(46,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,37)(34,41)(35,38)(36,46)(39,47)(40,42)(43,45)(44,48);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,20,21,29)(18,24,22,28)(19,27,25,31)(23,30,26,32)(33,38)(34,42)(35,37)(36,43)(39,48)(40,41)(44,47)(45,46)Polycyclic presentation:
2^9|a^2=i,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8973314);