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 | |
4055 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(7,14)(10,16)(13,15)(18,21)(23,25)(24,29)(30,31)(33,34,40,43)(35,44,37,38)(36,45,46,41)(39,47,48,42);(1,2)(3,8)(4,7)(5,9)(6,11)(10,15)(12,14)(13,16)(20,28)(24,29)(26,32)(30,31)(33,35,40,37)(34,38,43,44)(36,42,46,47)(39,45,48,41);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,18,22,21)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,36,40,46)(34,41,43,45)(35,42,37,47)(38,39,44,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,15)(9,14)(11,16)(17,19,22,27)(18,23,21,25)(20,26,28,32)(24,30,29,31)(33,37,40,35)(34,38,43,44)(36,42,46,47)(39,41,48,45);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,16)(13,14)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,38)(34,35)(36,39)(37,43)(40,44)(41,42)(45,47)(46,48);(1,2)(3,6)(4,7)(5,9)(8,11)(10,13)(12,14)(15,16)(17,21,22,18)(19,25,27,23)(20,29,28,24)(26,31,32,30)(33,39)(34,42)(35,45)(36,44)(37,41)(38,46)(40,48)(43,47)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,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, 9619987);