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 | |
4179 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,6,16)(4,10,13,5)(11,12,14,15)(17,18,22,24)(19,23,26,21)(20,25,30,32)(27,29,28,31)(33,34,40,39)(35,45,47,41)(36,42,37,43)(38,44,46,48);(1,3,7,6)(2,8,9,16)(4,11,13,14)(5,15,10,12)(17,19,22,26)(18,23,24,21)(20,28,30,27)(25,31,32,29)(33,35,40,47)(34,41,39,45)(36,46,37,38)(42,44,43,48);(2,9)(4,13)(8,16)(11,14)(18,24)(19,26)(25,32)(27,28)(33,36,40,37)(34,42,39,43)(35,46,47,38)(41,44,45,48);(1,4)(2,10)(3,11)(5,9)(6,14)(7,13)(8,12)(15,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,29)(28,31)(30,32)(33,37,40,36)(34,43,39,42)(35,38,47,46)(41,48,45,44);(1,5,7,10)(2,4,9,13)(3,12,6,15)(8,14,16,11)(17,20,22,30)(18,25,24,32)(19,27,26,28)(21,31,23,29)(33,38)(34,44)(35,36)(37,47)(39,48)(40,46)(41,43)(42,45);(1,6,7,3)(2,8,9,16)(4,14,13,11)(5,15,10,12)(17,21,22,23)(18,26,24,19)(20,29,30,31)(25,27,32,28)(33,39,40,34)(35,45,47,41)(36,43,37,42)(38,44,46,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9624635);