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 | |
4055 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,9,11,7)(5,8,13,12)(10,15,16,14)(17,18)(19,25)(20,28)(21,23)(22,27)(24,30)(26,31)(29,32)(33,34,39,43)(35,40,44,48)(36,37,41,47)(38,42,46,45);(2,4)(7,9)(8,12)(14,15)(17,19,21,27)(18,22,23,25)(20,29,30,26)(24,32,28,31)(33,35)(34,40)(36,45)(37,38)(39,44)(41,42)(43,48)(46,47);(5,13)(8,12)(10,16)(14,15)(18,23)(22,25)(24,28)(31,32)(35,44)(38,46)(40,48)(42,45);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,19)(18,22)(20,29)(21,27)(23,25)(24,32)(26,30)(28,31)(33,36)(34,37)(35,45)(38,40)(39,41)(42,44)(43,47)(46,48);(1,4,6,2)(3,7,11,9)(5,12,13,8)(10,14,16,15)(17,20)(18,24)(19,26)(21,30)(22,31)(23,28)(25,32)(27,29)(33,37)(34,41)(35,46)(36,43)(38,44)(39,47)(40,45)(42,48);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,18)(19,22)(20,24)(21,23)(25,27)(26,31)(28,30)(29,32)(33,38,39,46)(34,42,43,45)(35,37,44,47)(36,40,41,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,d^2=gh,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8281412);