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 | |
4086 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)(3,5)(4,12)(6,7)(8,10)(9,15)(11,16)(13,14)(17,18,23,20)(19,22,25,26)(21,24,30,28)(27,29,31,32)(33,34,39,40)(35,36,41,44)(37,45,47,38)(42,46,48,43);(2,7)(3,10)(9,12)(11,13)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,35,39,41)(34,36,40,44)(37,42,47,48)(38,43,45,46);(2,7)(5,8)(9,12)(14,16)(19,25)(22,26)(27,31)(29,32)(34,40)(35,41)(38,45)(42,48);(1,3)(2,8)(4,13)(5,7)(6,10)(9,14)(11,15)(12,16)(17,20,23,18)(19,26,25,22)(21,24,30,28)(27,29,31,32)(33,36,39,44)(34,41,40,35)(37,46,47,43)(38,42,45,48);(1,4,6,15)(2,9,7,12)(3,11,10,13)(5,14,8,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37)(34,38)(35,42)(36,43)(39,47)(40,45)(41,48)(44,46);(1,5,6,8)(2,3,7,10)(4,14,15,16)(9,11,12,13)(17,22,23,26)(18,25,20,19)(21,29,30,32)(24,31,28,27)(33,38)(34,37)(35,43)(36,42)(39,45)(40,47)(41,46)(44,48)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=i,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8241561);