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 | |
4241 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)(3,11)(4,14)(5,10)(6,7)(8,12)(9,15)(13,16)(33,34,38,41)(35,42,36,39)(37,40,45,48)(43,47,44,46);(2,7)(3,12)(9,14)(10,13)(17,18,22,23)(19,21,25,30)(20,24,29,31)(26,28,27,32)(35,36)(39,42)(43,44)(46,47);(2,7)(5,16)(8,11)(9,14)(18,23)(21,30)(24,31)(28,32)(33,35,38,36)(34,39,41,42)(37,44,45,43)(40,47,48,46);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,19,22,25)(18,21,23,30)(20,27,29,26)(24,32,31,28)(33,36,38,35)(34,39,41,42)(37,44,45,43)(40,46,48,47);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,20)(18,24)(19,26)(21,28)(22,29)(23,31)(25,27)(30,32)(33,34)(35,39)(36,42)(37,40)(38,41)(43,46)(44,47)(45,48);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,21)(18,25)(19,23)(20,28)(22,30)(24,27)(26,31)(29,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,48)(42,47)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8202256);