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 |
| prime radical | #36873 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4127 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)(3,11)(4,9)(5,10)(6,7)(8,12)(13,14)(15,16)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(33,34,39,43)(35,40,44,46)(36,41,45,48)(37,38,42,47);(2,7)(3,12)(9,14)(10,16)(18,24)(19,20)(26,29)(27,28)(35,44)(36,45)(40,46)(41,48);(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,35,39,44)(34,40,43,46)(36,38,45,47)(37,41,42,48);(1,3)(2,8)(4,10)(5,14)(6,12)(7,11)(9,15)(13,16)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,36)(34,41)(35,38)(37,46)(39,45)(40,42)(43,48)(44,47);(1,4)(2,9)(3,10)(5,11)(6,13)(7,14)(8,15)(12,16)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,37)(34,38)(35,41)(36,46)(39,42)(40,45)(43,47)(44,48);(1,5,6,15)(2,10,7,16)(3,9,12,14)(4,8,13,11)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,38)(34,42)(35,45)(36,44)(37,43)(39,47)(40,48)(41,46)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8603448);