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:
(1,2,6,7)(3,4,8,11)(5,9,13,14)(10,12,15,16)(17,18,22,24)(19,23,27,30)(20,21,28,29)(25,26,31,32)(33,34,38,39)(35,36,40,44)(37,45,46,41)(42,48,47,43);(2,7)(4,11)(9,14)(12,16)(17,19,22,27)(18,23,24,30)(20,25,28,31)(21,26,29,32)(33,35,38,40)(34,36,39,44)(37,42,46,47)(41,43,45,48);(5,13)(9,14)(10,15)(12,16)(18,24)(20,28)(23,30)(25,31)(34,39)(35,40)(37,46)(43,48);(1,3)(2,4)(5,10)(6,8)(7,11)(9,12)(13,15)(14,16)(17,20)(18,21)(19,25)(22,28)(23,26)(24,29)(27,31)(30,32)(33,34)(35,36)(37,41)(38,39)(40,44)(42,43)(45,46)(47,48);(1,4)(2,8)(3,7)(5,12)(6,11)(9,15)(10,14)(13,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,36)(34,40)(35,39)(37,43)(38,44)(41,47)(42,45)(46,48);(1,5,6,13)(2,9,7,14)(3,10,8,15)(4,12,11,16)(17,21)(18,20)(19,26)(22,29)(23,25)(24,28)(27,32)(30,31)(33,37)(34,41)(35,42)(36,43)(38,46)(39,45)(40,47)(44,48)Polycyclic presentation:
2^9|a^2=gh,b^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^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9385520);