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 | |
4270 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)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18,23,25)(19,22,20,24)(21,26,30,32)(27,29,28,31)(33,34,40,42)(35,43,37,41)(36,46,38,39)(44,45,47,48);(4,11)(8,14)(10,15)(12,16)(17,19,23,20)(18,24,25,22)(21,28,30,27)(26,29,32,31)(33,35,40,37)(34,41,42,43)(36,44,38,47)(39,45,46,48);(2,7)(3,9)(8,14)(10,15)(18,25)(19,20)(26,32)(27,28)(33,36,40,38)(34,39,42,46)(35,44,37,47)(41,45,43,48);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,29,32,31)(33,37,40,35)(34,41,42,43)(36,44,38,47)(39,48,46,45);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,23,30)(18,26,25,32)(19,27,20,28)(22,29,24,31)(33,38,40,36)(34,39,42,46)(35,44,37,47)(41,48,43,45);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22)(18,19)(20,25)(21,29)(23,24)(26,27)(28,32)(30,31)(33,39)(34,38)(35,45)(36,42)(37,48)(40,46)(41,47)(43,44)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gi,d^2=h,e^2=g,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, 9620661);