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 | |
4112 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)(3,8)(4,10)(5,15)(6,11)(7,9)(12,14)(13,16)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32);(1,3)(2,8)(4,9)(5,12)(6,13)(7,10)(11,16)(14,15)(18,19)(23,25)(24,26)(30,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,47)(44,45)(46,48);(2,9)(3,10)(5,11)(13,14)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,35,38,40)(34,37,41,42)(36,44,46,43)(39,47,48,45);(1,4,7,8)(2,10,9,3)(5,14,11,13)(6,16,15,12)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,47)(46,48);(1,5,7,11)(2,6,9,15)(3,12,10,16)(4,14,8,13)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36,38,46)(34,39,41,48)(35,43,40,44)(37,45,42,47);(1,6)(2,11)(3,13)(4,12)(5,9)(7,15)(8,16)(10,14)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37)(34,40)(35,41)(36,45)(38,42)(39,44)(43,48)(46,47)Polycyclic presentation:
2^9|c^2=ghi,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8441657);