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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3877 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)(3,7)(4,9)(5,10)(6,11)(8,12)(13,15)(14,16)(17,18,22,26)(19,21,25,23)(20,28,30,24)(27,29,32,31)(33,34)(35,43)(36,37)(38,40)(39,42)(41,47)(44,45)(46,48);(1,3)(2,7)(4,13)(5,14)(6,12)(8,11)(9,15)(10,16)(17,18,22,26)(19,21,25,23)(20,24,30,28)(27,31,32,29)(33,35,39,38)(34,40,42,43)(36,46,41,44)(37,45,47,48);(3,12)(7,8)(13,16)(14,15)(17,19,22,25)(18,23,26,21)(20,27,30,32)(24,29,28,31)(33,35,39,38)(34,40,42,43)(36,46,41,44)(37,45,47,48);(1,3)(2,8)(4,13)(5,15)(6,12)(7,11)(9,14)(10,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,29)(28,31)(30,32)(33,36,39,41)(34,37,42,47)(35,44,38,46)(40,48,43,45);(1,4)(2,9)(3,13)(5,11)(6,10)(7,15)(8,14)(12,16)(17,20,22,30)(18,24,26,28)(19,27,25,32)(21,31,23,29)(33,37)(34,41)(35,45)(36,42)(38,48)(39,47)(40,44)(43,46);(1,5,6,9)(2,10,11,4)(3,14,12,15)(7,16,8,13)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,27,28,32)(33,38,39,35)(34,40,42,43)(36,44,41,46)(37,45,47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8653775);