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 | |
4050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,23,21)(19,27,29,24)(20,30,31,25)(22,32,28,26)(33,34,39,45)(35,40,44,38)(36,43,47,41)(37,48,46,42);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19,23,29)(18,24,21,27)(20,28,31,22)(25,26,30,32)(33,35,39,44)(34,40,45,38)(36,37,47,46)(41,42,43,48);(2,6)(7,10)(8,12)(14,16)(17,20,23,31)(18,25,21,30)(19,22,29,28)(24,32,27,26)(33,36,39,47)(34,41,45,43)(35,46,44,37)(38,42,40,48);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,20)(18,25)(19,28)(21,30)(22,29)(23,31)(24,26)(27,32)(33,37,39,46)(34,42,45,48)(35,36,44,47)(38,43,40,41);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(17,21,23,18)(19,24,29,27)(20,30,31,25)(22,32,28,26)(33,36)(34,43)(35,46)(37,44)(38,48)(39,47)(40,42)(41,45);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22,23,28)(18,26,21,32)(19,20,29,31)(24,30,27,25)(33,38,39,40)(34,44,45,35)(36,42,47,48)(37,43,46,41)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9945682);