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 | |
4172 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)(4,8)(5,11)(6,9)(10,13)(12,15)(14,16)(17,18)(19,26)(20,23)(21,24)(22,28)(25,30)(27,31)(29,32)(33,34,38,41)(35,37,40,43)(36,39,46,48)(42,45,44,47);(1,3,6,11)(2,7,9,5)(4,10,14,12)(8,13,16,15)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35)(34,37)(36,44)(38,40)(39,47)(41,43)(42,46)(45,48);(1,3)(2,5)(4,10)(6,11)(7,9)(8,15)(12,14)(13,16)(17,18,23,20)(19,26,21,24)(22,25,30,28)(27,32,29,31)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,48)(43,47);(1,3)(2,5)(4,12)(6,11)(7,9)(8,13)(10,14)(15,16)(17,20,23,18)(19,26,21,24)(22,28,30,25)(27,32,29,31)(33,34)(35,43)(36,39)(37,40)(38,41)(42,47)(44,45)(46,48);(1,4,6,14)(2,8,9,16)(3,10,11,12)(5,15,7,13)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,37,38,43)(34,40,41,35)(36,45,46,47)(39,44,48,42);(1,5)(2,3)(4,13)(6,7)(8,12)(9,11)(10,16)(14,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,36)(34,39)(35,44)(37,47)(38,46)(40,42)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=g,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10411803);