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 |
| prime radical | #2617 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4056 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)(3,10)(4,12)(5,9)(6,7)(8,13)(11,14)(15,16)(17,18,22,25)(19,21,27,23)(20,24,29,31)(26,28,32,30)(33,34,38,42)(35,39,43,48)(36,40,46,45)(37,41,47,44);(2,7)(3,9)(4,13)(14,16)(17,19,22,27)(18,23,25,21)(20,26,29,32)(24,30,31,28)(35,43)(36,46)(39,48)(40,45);(2,7)(5,10)(8,12)(14,16)(17,20,22,29)(18,24,25,31)(19,26,27,32)(21,28,23,30)(33,35)(34,39)(36,44)(37,40)(38,43)(41,46)(42,48)(45,47);(1,3,6,9)(2,5,7,10)(4,11,13,15)(8,14,12,16)(17,21)(18,19)(20,28)(22,23)(24,26)(25,27)(29,30)(31,32);(1,4)(2,8)(3,11)(5,14)(6,13)(7,12)(9,15)(10,16)(17,19)(18,21)(20,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,36)(34,40)(35,44)(37,39)(38,46)(41,43)(42,45)(47,48);(1,5)(2,9)(3,7)(4,14)(6,10)(8,15)(11,12)(13,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37)(34,41)(35,45)(36,39)(38,47)(40,43)(42,44)(46,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109688);