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 | |
4077 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,5)(3,9,11,7)(4,8,12,13)(10,16,14,15)(17,18)(19,27)(20,31)(21,25)(22,26)(23,30)(24,29)(28,32)(35,37)(36,40)(43,45)(44,46);(1,3)(2,7)(4,10)(5,9)(6,11)(8,15)(12,14)(13,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,32)(25,31)(26,27)(33,34,39,42)(35,36,37,40)(38,41,47,48)(43,44,45,46);(4,12)(8,13)(10,14)(15,16)(17,18,22,26)(19,23,30,27)(20,24,29,31)(21,25,28,32)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,46,48,44);(1,4,6,12)(2,8,5,13)(3,10,11,14)(7,15,9,16)(17,18)(19,27)(20,24)(21,32)(22,26)(23,30)(25,28)(29,31)(33,36,39,40)(34,35,42,37)(38,46,47,44)(41,45,48,43);(1,5,6,2)(3,9,11,7)(4,13,12,8)(10,16,14,15)(17,20)(18,24)(19,28)(21,30)(22,29)(23,32)(25,27)(26,31)(33,37,39,35)(34,40,42,36)(38,43,47,45)(41,44,48,46);(1,3)(2,9)(4,14)(5,7)(6,11)(8,15)(10,12)(13,16)(17,21,22,28)(18,25,26,32)(19,29,30,20)(23,31,27,24)(33,38,39,47)(34,41,42,48)(35,43,37,45)(36,44,40,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=h,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8646519);