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 | |
4070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,11,13,7)(4,14,15,9)(5,10,16,12)(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);(1,3,6,13)(2,7,8,11)(4,10,15,12)(5,9,16,14)(17,19,22,27)(18,23,25,21)(20,26,29,32)(24,30,31,28)(35,43)(36,46)(39,48)(40,45);(2,8)(5,16)(7,11)(9,14)(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);(1,4,6,15)(2,9,8,14)(3,12,13,10)(5,11,16,7)(17,21)(18,19)(20,28)(22,23)(24,26)(25,27)(29,30)(31,32);(1,3)(2,7)(4,12)(5,9)(6,13)(8,11)(10,15)(14,16)(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)(2,10)(3,9)(4,11)(6,16)(7,15)(8,12)(13,14)(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=gi,b^2=g,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, 8109496);