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 | |
4107 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)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(33,34)(35,38)(36,41)(37,42)(39,44)(40,45)(43,46)(47,48);(2,6)(7,9)(8,12)(14,15)(18,22)(19,25)(24,29)(27,31)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(2,6)(4,13)(7,9)(10,16)(19,25)(20,28)(23,26)(24,29)(34,35)(37,45)(39,41)(43,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,35,38,34)(36,39,44,41)(37,42,45,40)(43,46,48,47);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,37)(34,40)(35,42)(36,43)(38,45)(39,46)(41,47)(44,48)Polycyclic presentation:
2^9|a^2=ghi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8841390);