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 | |
3968 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)(3,11)(4,5)(6,8)(7,9)(10,15)(12,13)(14,16)(17,18,22,26)(19,23,29,24)(20,21,25,31)(27,28,32,30)(33,34)(35,37)(36,44)(38,42)(39,40)(41,47)(43,46)(45,48);(1,3,7,6)(2,8,9,11)(4,14,10,12)(5,13,15,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,26)(25,32)(30,31)(34,40)(35,42)(41,44)(43,48);(2,9)(4,10)(8,11)(12,14)(17,19,22,29)(18,23,26,24)(20,27,25,32)(21,28,31,30)(33,35,39,42)(34,37,40,38)(36,43,47,48)(41,45,44,46);(1,4,7,10)(2,5,9,15)(3,12,6,14)(8,16,11,13)(17,19)(18,24)(20,27)(21,30)(22,29)(23,26)(25,32)(28,31)(33,36,39,47)(34,41,40,44)(35,43,42,48)(37,45,38,46);(1,5)(2,10)(3,13)(4,9)(6,16)(7,15)(8,12)(11,14)(17,20)(18,21)(19,27)(22,25)(23,28)(24,30)(26,31)(29,32)(33,37,39,38)(34,35,40,42)(36,45,47,46)(41,43,44,48);(1,6,7,3)(2,8,9,11)(4,12,10,14)(5,13,15,16)(17,21)(18,25)(19,28)(20,26)(22,31)(23,32)(24,27)(29,30)(33,38,39,37)(34,42,40,35)(36,46,47,45)(41,48,44,43)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=i,d^2=g,e^2=gh,f^2=g,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, 8649160);