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 | |
4250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,4,8)(5,13,14,10)(11,15,12,16)(17,18,23,27)(19,26,30,24)(20,22,21,25)(28,29,31,32)(33,34,38,36)(35,41,43,39)(37,44,45,40)(42,46,48,47);(2,7)(8,9)(10,13)(15,16)(17,19)(18,24)(20,31)(21,28)(22,29)(23,30)(25,32)(26,27)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(1,3,6,4)(2,8,7,9)(5,11,14,12)(10,15,13,16)(17,20,23,21)(18,25,27,22)(19,28,30,31)(24,29,26,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,4,6,3)(2,9,7,8)(5,11,14,12)(10,15,13,16)(17,21,23,20)(18,22,27,25)(19,28,30,31)(24,29,26,32)(33,36,38,34)(35,41,43,39)(37,40,45,44)(42,46,48,47);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,19)(18,26)(20,28)(21,31)(22,29)(23,30)(24,27)(25,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,22)(18,20)(19,29)(21,27)(23,25)(24,31)(26,28)(30,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=h,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, 10315758);