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 | |
4252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(4,11)(15,16)(18,23)(19,25)(21,29)(28,31)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,44)(47,48);(1,2,5,6)(3,7,9,13)(4,8,11,14)(10,15,12,16)(17,18,22,23)(19,20,25,24)(21,26,29,30)(27,28,32,31);(1,3)(2,7)(4,10)(5,9)(6,13)(8,15)(11,12)(14,16)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(1,3)(2,7)(4,12)(5,9)(6,13)(8,16)(10,11)(14,15)(17,19)(18,20)(21,27)(22,25)(23,24)(26,28)(29,32)(30,31)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,15)(11,14)(12,16)(17,20)(18,25)(19,23)(21,28)(22,24)(26,32)(27,30)(29,31)(33,36)(34,39)(35,41)(37,44)(38,42)(40,47)(43,45)(46,48);(1,4)(2,8)(3,10)(5,11)(6,14)(7,15)(9,12)(13,16)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)Polycyclic presentation:
2^9|b^2=g,c^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10315321);