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 | #57157 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4145 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)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(17,18,23,27)(19,24,30,32)(20,22,31,25)(21,29,28,26)(33,34,40,45)(35,43,37,41)(36,42,38,47)(39,44,48,46);(2,5)(7,9)(8,12)(14,15)(17,19,23,30)(18,24,27,32)(20,28,31,21)(22,26,25,29)(33,35,40,37)(34,41,45,43)(36,46,38,44)(39,42,48,47);(1,3,6,11)(2,7,5,9)(4,10,13,16)(8,14,12,15)(17,20,23,31)(18,25,27,22)(19,21,30,28)(24,26,32,29)(33,36,40,38)(34,42,45,47)(35,46,37,44)(39,41,48,43);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,15)(13,16)(17,20)(18,22)(19,28)(21,30)(23,31)(24,26)(25,27)(29,32)(33,37,40,35)(34,43,45,41)(36,46,38,44)(39,42,48,47);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21,23,28)(18,26,27,29)(19,20,30,31)(22,24,25,32)(33,38,40,36)(34,42,45,47)(35,46,37,44)(39,43,48,41);(1,5,6,2)(3,9,11,7)(4,12,13,8)(10,15,16,14)(17,22)(18,20)(19,29)(21,24)(23,25)(26,30)(27,31)(28,32)(33,39,40,48)(34,44,45,46)(35,47,37,42)(36,41,38,43)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gi,d^2=ghi,e^2=gh,f^2=g,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, 8662516);