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 | |
4100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,13,8)(4,5,9,15)(11,12,16,14)(17,18,23,27)(19,25,21,24)(20,26,31,22)(28,29,30,32)(33,34)(35,43)(36,37)(38,41)(39,40)(42,47)(44,45)(46,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,10)(9,16)(14,15)(17,19,23,21)(18,24,27,25)(20,28,31,30)(22,29,26,32)(34,40)(36,42)(41,43)(44,46);(1,3,6,13)(2,7,10,8)(4,11,9,16)(5,12,15,14)(17,19,23,21)(18,25,27,24)(20,30,31,28)(22,29,26,32)(33,35,39,38)(34,41,40,43)(36,46,42,44)(37,45,47,48);(1,3)(2,8)(4,11)(5,14)(6,13)(7,10)(9,16)(12,15)(17,20,23,31)(18,26,27,22)(19,28,21,30)(24,32,25,29)(33,36,39,42)(34,37,40,47)(35,44,38,46)(41,48,43,45);(1,4)(2,5)(3,11)(6,9)(7,12)(8,14)(10,15)(13,16)(17,21,23,19)(18,24,27,25)(20,30,31,28)(22,29,26,32)(33,37)(34,42)(35,45)(36,40)(38,48)(39,47)(41,44)(43,46);(1,5)(2,9)(3,12)(4,10)(6,15)(7,16)(8,11)(13,14)(17,22,23,26)(18,20,27,31)(19,29,21,32)(24,28,25,30)(33,38,39,35)(34,41,40,43)(36,44,42,46)(37,45,47,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gi,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, 8649166);