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 | |
4074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,11)(4,9)(5,13)(6,7)(8,12)(10,15)(14,16)(17,18,23,25)(19,24,28,32)(20,21,26,31)(22,30,29,27)(33,34,38,41)(35,37,43,39)(36,40,44,47)(42,45,48,46);(2,7)(4,14)(8,11)(10,13)(17,19,23,28)(18,24,25,32)(20,29,26,22)(21,27,31,30)(36,44)(40,47)(42,48)(45,46);(2,7)(3,12)(9,16)(10,13)(18,25)(19,28)(20,26)(27,30)(33,35,38,43)(34,39,41,37)(36,42,44,48)(40,46,47,45);(1,3)(2,8)(4,10)(5,9)(6,12)(7,11)(13,14)(15,16)(17,20)(18,21)(19,29)(22,28)(23,26)(24,27)(25,31)(30,32)(33,35)(34,37)(36,42)(38,43)(39,41)(40,45)(44,48)(46,47);(1,4,6,14)(2,9,7,16)(3,13,12,10)(5,8,15,11)(17,21)(18,26)(19,30)(20,25)(22,32)(23,31)(24,29)(27,28)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,5,6,15)(2,10,7,13)(3,9,12,16)(4,11,14,8)(17,22,23,29)(18,27,25,30)(19,20,28,26)(21,24,31,32)(33,37)(34,35)(36,45)(38,39)(40,42)(41,43)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=ghi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10103310);