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 | |
4270 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)(3,5)(4,12)(6,7)(8,10)(9,13)(11,16)(14,15)(19,25)(23,26)(27,31)(30,32)(33,34)(35,37)(36,44)(38,39)(40,43)(41,47)(42,48)(45,46);(2,7)(4,13)(5,8)(11,14)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(34,39)(37,40)(41,44)(46,48);(2,7)(3,10)(9,12)(11,14)(18,22)(23,26)(24,28)(30,32)(33,35,38,43)(34,37,39,40)(36,42,47,45)(41,46,44,48);(1,3,6,10)(2,8,7,5)(4,14,13,11)(9,15,12,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35)(34,40)(36,45)(37,39)(38,43)(41,46)(42,47)(44,48);(1,4,6,13)(2,9,7,12)(3,11,10,14)(5,15,8,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,36,38,47)(34,41,39,44)(35,42,43,45)(37,46,40,48);(1,5,6,8)(2,10,7,3)(4,15,13,16)(9,14,12,11)(17,20,21,29)(18,24,22,28)(19,27,25,31)(23,30,26,32)(33,37,38,40)(34,35,39,43)(36,46,47,48)(41,42,44,45)Polycyclic presentation:
2^9|c^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8782785);