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 | |
4120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,7,11,4)(5,8,14,16)(10,13,15,12)(17,18)(19,24)(20,22)(21,28)(23,27)(25,29)(26,30)(31,32)(33,34)(35,38)(36,45)(37,41)(39,43)(40,42)(44,48)(46,47);(1,3,6,11)(2,4,9,7)(5,10,14,15)(8,12,16,13)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,35)(34,40)(36,44)(37,47)(38,43)(39,42)(41,48)(45,46);(17,18,22,20)(19,24,27,23)(21,25,29,28)(26,31,32,30)(33,36,39,37)(34,41,43,45)(35,44,42,47)(38,46,40,48);(1,3)(2,7)(4,9)(5,10)(6,11)(8,13)(12,16)(14,15)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,37,39,36)(34,41,43,45)(35,44,42,47)(38,48,40,46);(1,4,6,7)(2,3,9,11)(5,13,14,12)(8,15,16,10)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,35)(34,38)(36,44)(37,47)(39,42)(40,43)(41,46)(45,48);(1,5,6,14)(2,8,9,16)(3,10,11,15)(4,12,7,13)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,38)(34,42)(35,43)(36,46)(37,48)(39,40)(41,47)(44,45)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10362904);