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 | |
4159 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)(3,7)(4,9)(5,10)(6,11)(8,13)(12,15)(14,16)(17,18,22,25)(19,23,28,31)(20,21,29,24)(26,27,32,30)(33,34)(35,47)(36,42)(37,43)(38,44)(39,45)(40,46)(41,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,11)(12,14)(15,16)(17,19,22,28)(18,23,25,31)(20,26,29,32)(21,27,24,30)(33,35)(34,41)(36,43)(37,44)(38,39)(40,48)(42,45)(46,47);(1,3,6,13)(2,7,11,8)(4,12,14,10)(5,9,15,16)(17,20,22,29)(18,24,25,21)(19,26,28,32)(23,30,31,27)(33,36,40,38)(34,42,46,44)(35,43,48,39)(37,41,45,47);(1,3)(2,8)(4,12)(5,9)(6,13)(7,11)(10,14)(15,16)(17,20)(18,21)(19,26)(22,29)(23,27)(24,25)(28,32)(30,31)(33,37,40,45)(34,43,46,39)(35,44,48,42)(36,47,38,41);(1,4)(2,9)(3,12)(5,8)(6,14)(7,15)(10,13)(11,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,38,40,36)(34,44,46,42)(35,43,48,39)(37,41,45,47);(1,5)(2,10)(3,9)(4,7)(6,15)(8,14)(11,12)(13,16)(17,21)(18,20)(19,27)(22,24)(23,26)(25,29)(28,30)(31,32)(33,39,40,43)(34,45,46,37)(35,38,48,36)(41,42,47,44)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=ghi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dghi,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9877328);