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 | |
4122 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)(4,5)(6,7)(8,13)(9,14)(11,12)(15,16)(33,34,39,43)(35,42,37,40)(36,41,46,38)(44,48,47,45);(2,7)(4,14)(8,10)(11,15)(17,18,22,23)(19,25,28,24)(20,29,31,21)(26,32,30,27)(35,37)(40,42)(44,47)(45,48);(2,7)(5,9)(8,10)(12,16)(18,23)(21,29)(24,25)(27,32)(33,35,39,37)(34,40,43,42)(36,44,46,47)(38,48,41,45);(1,3,6,13)(2,8,7,10)(4,15,14,11)(5,12,9,16)(17,19,22,28)(18,24,23,25)(20,30,31,26)(21,27,29,32)(33,36)(34,41)(35,44)(37,47)(38,43)(39,46)(40,45)(42,48);(1,4)(2,9)(3,11)(5,7)(6,14)(8,12)(10,16)(13,15)(17,20)(18,21)(19,26)(22,31)(23,29)(24,32)(25,27)(28,30)(33,37,39,35)(34,42,43,40)(36,44,46,47)(38,48,41,45);(1,5,6,9)(2,4,7,14)(3,12,13,16)(8,15,10,11)(17,21)(18,20)(19,27)(22,29)(23,31)(24,30)(25,26)(28,32)(33,38,39,41)(34,36,43,46)(35,45,37,48)(40,47,42,44)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8302215);