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 | |
4206 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)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,22)(20,23)(21,26)(24,29)(25,28)(27,31)(30,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46);(3,9)(7,13)(10,15)(14,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(2,6)(7,13)(8,11)(14,16)(18,19)(23,25)(24,26)(30,31);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,47)(44,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37,38,46)(34,40,41,45)(35,43,36,44)(39,47,42,48)Polycyclic presentation:
2^9|b^2=g,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9817937);