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 | |
4005 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,8)(4,5,9,15)(12,13,16,14)(17,18,23,20)(19,24,21,25)(22,26,30,28)(27,31,29,32)(33,34)(35,44)(36,37)(38,42)(39,43)(40,45)(41,47)(46,48);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,16)(14,15)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,35,39,45)(34,40,43,44)(36,46,41,38)(37,42,47,48);(3,11)(7,8)(12,16)(13,14)(17,19,23,21)(18,25,20,24)(22,27,30,29)(26,32,28,31)(33,35,39,45)(34,40,43,44)(36,46,41,38)(37,42,47,48);(1,3)(2,8)(4,12)(5,14)(6,11)(7,10)(9,16)(13,15)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32)(33,36,39,41)(34,37,43,47)(35,38,45,46)(40,48,44,42);(1,4)(2,5)(3,12)(6,9)(7,13)(8,14)(10,15)(11,16)(17,21,23,19)(18,24,20,25)(22,29,30,27)(26,31,28,32)(33,37)(34,41)(35,42)(36,43)(38,40)(39,47)(44,46)(45,48);(1,5)(2,9)(3,13)(4,10)(6,15)(7,16)(8,12)(11,14)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)(33,38)(34,42)(35,36)(37,44)(39,46)(40,47)(41,45)(43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=h,d^2=h,e^2=gh,f^2=gh,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, 8641167);