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 | |
4160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(7,9)(8,13)(14,15)(17,18)(19,27)(20,28)(21,26)(22,24)(23,29)(25,31)(30,32)(33,34)(35,39)(36,41)(37,45)(38,40)(42,48)(43,44)(46,47);(1,2,5,6)(3,9,11,7)(4,8,12,13)(10,15,16,14)(17,19,22,29)(18,23,24,27)(20,26,31,30)(21,28,32,25)(33,35,38,41)(34,39,40,36)(37,46,48,43)(42,44,45,47);(4,12)(8,13)(10,16)(14,15)(18,24)(21,32)(23,27)(25,28)(34,40)(35,41)(42,45)(43,46);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20)(18,25)(19,26)(21,23)(22,31)(24,28)(27,32)(29,30)(33,36,38,39)(34,41,40,35)(37,47,48,44)(42,43,45,46);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,19)(18,23)(20,30)(21,28)(22,29)(24,27)(25,32)(26,31)(33,37,38,48)(34,42,40,45)(35,43,41,46)(36,44,39,47);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,21,22,32)(18,26,24,30)(19,28,29,25)(20,23,31,27)(33,34)(35,39)(36,41)(37,42)(38,40)(43,47)(44,46)(45,48)Polycyclic presentation:
2^9|b^2=gi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8300561);