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 | |
3982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,11,9)(4,12,13,8)(10,15,16,14)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(33,34,39,42)(35,37,40,36)(38,41,46,48)(43,45,47,44);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(18,22)(23,25)(24,28)(30,31)(33,35,39,40)(34,36,42,37)(38,43,46,47)(41,44,48,45);(2,6)(7,9)(8,12)(14,15)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,36,39,37)(34,35,42,40)(38,44,46,45)(41,43,48,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,15)(13,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,31)(30,32)(33,37,39,36)(34,40,42,35)(38,44,46,45)(41,43,48,47);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,38,39,46)(34,41,42,48)(35,43,40,47)(36,44,37,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9592777);