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 | |
4147 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,8,6)(3,5,9,4)(7,14,15,10)(11,12,16,13)(17,18,23,28)(19,24,30,32)(20,21,26,25)(22,31,29,27)(34,40)(35,36)(38,42)(45,47);(1,3)(2,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,16)(17,19,23,30)(18,24,28,32)(20,29,26,22)(21,27,25,31)(33,34)(35,44)(36,41)(37,42)(38,43)(39,40)(45,46)(47,48);(1,4,8,5)(2,9,6,3)(7,12,15,13)(10,16,14,11)(17,20)(18,25)(19,29)(21,28)(22,30)(23,26)(24,31)(27,32)(33,35,39,36)(34,41,40,44)(37,45,43,47)(38,46,42,48);(1,5,8,4)(2,9,6,3)(7,13,15,12)(10,16,14,11)(17,20)(18,21)(19,29)(22,30)(23,26)(24,27)(25,28)(31,32)(33,36,39,35)(34,41,40,44)(37,45,43,47)(38,48,42,46);(1,6,8,2)(3,5,9,4)(7,10,15,14)(11,12,16,13)(17,21)(18,26)(19,27)(20,28)(22,24)(23,25)(29,32)(30,31)(33,37,39,43)(34,42,40,38)(35,45,36,47)(41,48,44,46);(1,7,8,15)(2,10,6,14)(3,11,9,16)(4,12,5,13)(17,22,23,29)(18,27,28,31)(19,20,30,26)(21,24,25,32)(33,38,39,42)(34,43,40,37)(35,46,36,48)(41,47,44,45)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=g,d^2=g,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^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10365098);