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 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,23,20)(19,26,22,24)(21,25,29,28)(27,32,30,31)(33,34,39,42)(35,43,46,37)(36,41,38,40)(44,47,45,48);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(21,29)(25,28)(27,30)(31,32)(33,35,39,46)(34,37,42,43)(36,44,38,45)(40,47,41,48);(2,6)(7,10)(8,12)(14,16)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32)(33,35)(34,37)(36,44)(38,45)(39,46)(40,47)(41,48)(42,43);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,23,18)(19,26,22,24)(21,28,29,25)(27,32,30,31)(33,36,39,38)(34,40,42,41)(35,44,46,45)(37,47,43,48);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,37)(34,35)(36,47)(38,48)(39,43)(40,44)(41,45)(42,46);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22,23,19)(18,26,20,24)(21,30,29,27)(25,32,28,31)(33,38,39,36)(34,41,42,40)(35,45,46,44)(37,48,43,47)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8663191);