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 | |
4163 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)(3,12)(4,5)(6,14)(7,11)(8,16)(9,15)(10,13)(20,28)(24,31)(26,32)(29,30)(34,40)(35,36)(37,47)(38,48);(1,3)(2,8)(4,13)(5,14)(6,15)(7,16)(9,10)(11,12)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32)(33,34,39,40)(35,41,36,46)(37,42,47,45)(38,43,48,44);(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,35,39,36)(34,41,40,46)(37,44,47,43)(38,45,48,42);(1,4,7,15)(2,9,11,5)(3,13,16,6)(8,10,12,14)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36,39,35)(34,41,40,46)(37,44,47,43)(38,42,48,45);(1,5,7,9)(2,4,11,15)(3,14,16,10)(6,8,13,12)(17,18)(19,22)(20,24)(21,25)(23,27)(26,29)(28,31)(30,32)(33,37)(34,42)(35,44)(36,43)(38,41)(39,47)(40,45)(46,48);(1,6,7,13)(2,10,11,14)(3,15,16,4)(5,8,9,12)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,38)(34,43)(35,45)(36,42)(37,41)(39,48)(40,44)(46,47)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044581);