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 | |
4228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(10,16)(14,15)(18,23)(19,27)(21,25)(28,30)(33,34)(35,41)(36,37)(38,46)(39,40)(42,47)(43,44)(45,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,29)(30,32)(34,40)(36,41)(42,46)(44,48);(2,6)(7,9)(8,13)(14,15)(19,27)(24,31)(28,30)(29,32)(33,35,39,37)(34,36,40,41)(38,43,47,45)(42,44,46,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,22,27)(18,24,23,31)(20,30,26,28)(21,29,25,32)(33,36)(34,35)(37,40)(38,44)(39,41)(42,43)(45,46)(47,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20,22,26)(18,25,23,21)(19,28,27,30)(24,29,31,32)(33,37,39,35)(34,41,40,36)(38,45,47,43)(42,48,46,44);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,25)(18,26,23,20)(19,29,27,32)(24,28,31,30)(33,38,39,47)(34,42,40,46)(35,43,37,45)(36,44,41,48)Polycyclic presentation:
2^9|c^2=h,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8062249);