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 | |
4102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,11,14,8)(4,15,13,9)(5,16,12,10)(20,28)(24,31)(27,32)(29,30)(34,38)(39,45)(40,46)(47,48);(1,3)(2,8)(4,12)(5,13)(6,11)(7,14)(9,10)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(34,38)(35,41)(40,46)(42,43);(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,34)(35,39)(36,40)(37,38)(41,45)(42,47)(43,48)(44,46);(1,4,7,13)(2,9,6,15)(3,12,14,5)(8,10,11,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,35)(34,39)(36,43)(37,41)(38,45)(40,48)(42,44)(46,47);(1,5,7,12)(2,10,6,16)(3,13,14,4)(8,9,11,15)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30);(1,6,7,2)(3,11,14,8)(4,9,13,15)(5,16,12,10)(17,21,22,23)(18,25,26,19)(20,29,28,30)(24,32,31,27)(33,36,37,44)(34,40,38,46)(35,42,41,43)(39,47,45,48)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8365922);