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 | |
4266 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)(3,7)(4,9)(5,10)(6,11)(8,12)(13,15)(14,16)(17,18,23,25)(19,27,21,24)(20,26,31,22)(28,32,30,29)(34,38)(35,41)(40,43)(42,44);(1,3)(2,7)(4,13)(5,14)(6,12)(8,11)(9,15)(10,16)(17,19,23,21)(18,24,25,27)(20,30,31,28)(22,29,26,32)(34,38)(39,46)(40,43)(47,48);(3,12)(7,8)(13,16)(14,15)(18,25)(20,31)(24,27)(28,30)(33,34,37,38)(35,39,41,46)(36,40,45,43)(42,47,44,48);(1,3)(2,8)(4,13)(5,15)(6,12)(7,11)(9,14)(10,16)(17,20,23,31)(18,26,25,22)(19,28,21,30)(24,29,27,32)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(1,4)(2,9)(3,13)(5,11)(6,10)(7,15)(8,14)(12,16)(17,21,23,19)(18,27,25,24)(20,30,31,28)(22,29,26,32)(33,35)(34,39)(36,44)(37,41)(38,46)(40,48)(42,45)(43,47);(1,5,6,9)(2,10,11,4)(3,14,12,15)(7,16,8,13)(17,22,23,26)(18,20,25,31)(19,29,21,32)(24,30,27,28)(33,36,37,45)(34,40,38,43)(35,42,41,44)(39,47,46,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=g,e^2=g,f^2=gi,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, 8653446);