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 | |
4100 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)(17,18,23,29)(19,24,28,22)(20,27,31,25)(21,32,30,26)(33,34)(35,43)(36,46)(37,40)(38,42)(39,41)(44,47)(45,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19,23,28)(18,24,29,22)(20,21,31,30)(25,26,27,32)(33,35,40,39)(34,41,37,43)(36,45,38,44)(42,48,46,47);(2,6)(7,10)(8,12)(14,16)(17,20,23,31)(18,25,29,27)(19,30,28,21)(22,26,24,32)(33,36,40,38)(34,42,37,46)(35,44,39,45)(41,47,43,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,21,23,30)(18,26,29,32)(19,20,28,31)(22,27,24,25)(33,37,40,34)(35,43,39,41)(36,42,38,46)(44,47,45,48);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(17,20)(18,27)(19,30)(21,28)(22,32)(23,31)(24,26)(25,29)(33,38,40,36)(34,42,37,46)(35,45,39,44)(41,47,43,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22,23,24)(18,28,29,19)(20,26,31,32)(21,27,30,25)(33,39,40,35)(34,43,37,41)(36,45,38,44)(42,48,46,47)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,d^2=h,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, 9930551);