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 | |
3954 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,23)(20,27)(21,29)(22,25)(24,30)(26,28)(31,32)(33,34,40,39)(35,45,47,41)(36,42,37,43)(38,44,46,48);(3,9)(7,13)(10,15)(14,16)(17,19,22,27)(18,23,25,20)(21,26,30,32)(24,31,29,28)(33,35)(34,41)(36,38)(37,46)(39,45)(40,47)(42,48)(43,44);(2,6)(7,13)(8,11)(14,16)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,29,32)(33,36,40,37)(34,42,39,43)(35,38,47,46)(41,48,45,44);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31)(33,37,40,36)(34,43,39,42)(35,46,47,38)(41,44,45,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,18)(19,20)(21,29)(22,25)(23,27)(24,30)(26,31)(28,32)(33,38,40,46)(34,44,39,48)(35,36,47,37)(41,43,45,42);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(33,39,40,34)(35,45,47,41)(36,43,37,42)(38,44,46,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gi,d^2=g,e^2=g,f^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, 9634829);