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 | |
3975 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(18,23)(19,21)(20,28)(31,32)(34,39)(35,42)(36,44)(45,47);(2,6)(7,9)(8,12)(14,15)(17,18,22,23)(19,24,21,26)(20,29,28,25)(27,32,30,31)(33,34,38,39)(35,37,42,40)(36,41,44,46)(43,45,48,47);(3,10)(7,9)(11,16)(14,15)(17,19,22,21)(18,24,23,26)(20,30,28,27)(25,32,29,31)(33,35,38,42)(34,40,39,37)(36,43,44,48)(41,47,46,45);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,20,22,28)(18,25,23,29)(19,27,21,30)(24,31,26,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,29)(30,32)(33,36,38,44)(34,41,39,46)(35,43,42,48)(37,45,40,47);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21,22,19)(18,26,23,24)(20,30,28,27)(25,32,29,31)(33,37,38,40)(34,42,39,35)(36,45,44,47)(41,48,46,43)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8021584);