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 | |
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:
(1,2,7,12)(3,8,14,16)(4,5,10,9)(6,15,13,11)(17,18)(19,26)(20,24)(21,29)(22,23)(25,30)(27,31)(28,32)(33,34,38,41)(35,42,36,39)(37,45,46,40)(43,47,44,48);(1,3,7,14)(2,8,12,16)(4,13,10,6)(5,11,9,15)(18,23)(24,26)(25,29)(31,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46);(1,4)(2,9)(3,13)(5,12)(6,14)(7,10)(8,15)(11,16)(17,19,22,20)(18,24,23,26)(21,27,30,28)(25,31,29,32)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(1,4)(2,5)(3,13)(6,14)(7,10)(8,11)(9,12)(15,16)(17,18)(19,24)(20,26)(21,25)(22,23)(27,31)(28,32)(29,30)(33,34)(35,39)(36,42)(37,40)(38,41)(43,47)(44,48)(45,46);(1,5)(2,10)(3,11)(4,12)(6,8)(7,9)(13,16)(14,15)(17,20,22,19)(18,24,23,26)(21,27,30,28)(25,32,29,31)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,6,7,13)(2,11,12,15)(3,4,14,10)(5,8,9,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10365120);