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 | |
3947 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)(3,9,14,7)(4,5,15,8)(11,16,12,13)(17,18,23,27)(19,25,21,24)(20,26,31,22)(28,29,30,32)(33,34)(35,37)(36,44)(38,42)(39,40)(41,47)(43,46)(45,48);(1,3)(2,7)(4,11)(5,13)(6,14)(8,16)(9,10)(12,15)(17,19,23,21)(18,24,27,25)(20,28,31,30)(22,29,26,32)(34,40)(35,42)(41,44)(43,48);(1,4,6,15)(2,8,10,5)(3,11,14,12)(7,16,9,13)(17,19,23,21)(18,25,27,24)(20,30,31,28)(22,29,26,32)(33,35,39,42)(34,37,40,38)(36,43,47,48)(41,45,44,46);(1,4)(2,5)(3,12)(6,15)(7,16)(8,10)(9,13)(11,14)(17,20,23,31)(18,26,27,22)(19,28,21,30)(24,32,25,29)(33,36,39,47)(34,41,40,44)(35,43,42,48)(37,45,38,46);(1,5)(2,4)(3,13)(6,8)(7,11)(9,12)(10,15)(14,16)(17,21,23,19)(18,24,27,25)(20,30,31,28)(22,29,26,32)(33,37,39,38)(34,35,40,42)(36,45,47,46)(41,43,44,48);(1,3)(2,9)(4,11)(5,16)(6,14)(7,10)(8,13)(12,15)(17,22,23,26)(18,20,27,31)(19,29,21,32)(24,28,25,30)(33,38,39,37)(34,42,40,35)(36,46,47,45)(41,48,44,43)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=g,d^2=gh,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, 8646407);