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 | |
4169 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,10,4)(5,9,14,15)(11,13,16,12)(17,18,22,25)(19,26,21,23)(20,24,28,31)(27,32,29,30)(34,38)(35,41)(40,43)(42,47);(1,3)(2,7)(4,8)(5,11)(6,10)(9,13)(12,15)(14,16)(17,19,22,21)(18,23,25,26)(20,27,28,29)(24,30,31,32)(33,34,37,38)(35,39,41,45)(36,40,44,43)(42,46,47,48);(2,8)(3,10)(9,15)(11,16)(20,28)(24,31)(27,29)(30,32)(33,35,37,41)(34,39,38,45)(36,42,44,47)(40,46,43,48);(1,4)(2,3)(5,13)(6,7)(8,10)(9,16)(11,15)(12,14)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,34)(35,39)(36,43)(37,38)(40,44)(41,45)(42,48)(46,47);(1,3)(2,4)(5,11)(6,10)(7,8)(9,12)(13,15)(14,16)(17,20,22,28)(18,24,25,31)(19,27,21,29)(23,30,26,32)(33,35)(34,39)(36,42)(37,41)(38,45)(40,46)(43,48)(44,47);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,21,22,19)(18,23,25,26)(20,29,28,27)(24,30,31,32)(33,36,37,44)(34,40,38,43)(35,42,41,47)(39,46,45,48)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=hi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8442248);