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 | #36853 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,10,14)(4,8,13,11)(9,15,12,16)(17,18)(19,26)(20,23)(21,28)(22,25)(24,30)(27,29)(31,32)(33,34,39,43)(35,40,44,46)(36,41,45,48)(37,38,42,47);(2,6)(7,14)(8,11)(15,16)(18,20)(19,25)(24,28)(27,31)(35,44)(36,45)(40,46)(41,48);(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,35,39,44)(34,40,43,46)(36,38,45,47)(37,41,42,48);(1,2)(3,7)(4,11)(5,6)(8,13)(9,16)(10,14)(12,15)(17,20,23,18)(19,26,25,22)(21,28,30,24)(27,32,31,29)(33,36)(34,41)(35,38)(37,46)(39,45)(40,42)(43,48)(44,47);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,21,23,30)(18,24,20,28)(19,27,25,31)(22,29,26,32)(33,37)(34,38)(35,41)(36,46)(39,42)(40,45)(43,47)(44,48);(1,4,5,13)(2,8,6,11)(3,9,10,12)(7,15,14,16)(17,22,23,26)(18,25,20,19)(21,29,30,32)(24,31,28,27)(33,38)(34,42)(35,45)(36,44)(37,43)(39,47)(40,48)(41,46)Polycyclic presentation:
2^9|a^2=gi,c^2=hi,d^2=h,e^2=g,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, 8603013);