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 | |
4178 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,15)(13,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,34,39,41)(35,40,44,48)(36,38,43,46)(37,45,47,42);(2,7)(3,11)(4,14)(13,15)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,35,39,44)(34,40,41,48)(36,42,43,45)(37,46,47,38);(2,7)(5,9)(8,10)(13,15)(20,28)(23,30)(26,32)(29,31)(34,41)(35,44)(38,46)(42,45);(1,3,6,11)(2,8,7,10)(4,12,14,16)(5,13,9,15)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,36,39,43)(34,38,41,46)(35,45,44,42)(37,40,47,48);(1,4)(2,9)(3,12)(5,7)(6,14)(8,15)(10,13)(11,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,37)(34,42)(35,38)(36,40)(39,47)(41,45)(43,48)(44,46);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,12,10,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,38)(34,43)(35,37)(36,41)(39,46)(40,45)(42,48)(44,47)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109395);