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 | |
4427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,6,4,8)(5,10,14,16)(11,13,12,15)(17,18,22,20)(19,25,27,23)(21,24,30,28)(26,32,29,31)(33,34)(35,37)(36,40)(38,41)(39,42)(43,45)(44,46)(47,48);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,35,39,36)(34,40,42,37)(38,43,47,44)(41,46,48,45);(2,9)(3,4)(10,16)(11,12)(18,20)(23,25)(24,28)(31,32)(33,35,39,36)(34,37,42,40)(38,43,47,44)(41,45,48,46);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,20,22,18)(19,25,27,23)(21,28,30,24)(26,32,29,31)(33,36,39,35)(34,40,42,37)(38,44,47,43)(41,46,48,45);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,13,8,15)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,37,39,40)(34,35,42,36)(38,46,47,45)(41,44,48,43);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,46)(42,48)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9820486);