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 | |
4244 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,8)(4,5,9,15)(12,13,16,14)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,31)(29,30)(33,34,39,43)(35,44,46,40)(36,42,47,41)(37,48,45,38);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,16)(14,15)(17,19,22,21)(18,23,25,26)(20,27,28,29)(24,30,31,32)(33,35)(34,40)(36,37)(38,42)(39,46)(41,48)(43,44)(45,47);(3,11)(7,8)(12,16)(13,14)(20,28)(24,31)(27,29)(30,32)(33,36,39,47)(34,41,43,42)(35,37,46,45)(38,40,48,44);(1,3)(2,8)(4,12)(5,14)(6,11)(7,10)(9,16)(13,15)(17,20,22,28)(18,24,25,31)(19,27,21,29)(23,30,26,32)(33,36)(34,42)(35,45)(37,46)(38,44)(39,47)(40,48)(41,43);(1,4)(2,5)(3,12)(6,9)(7,13)(8,14)(10,15)(11,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,37,39,45)(34,38,43,48)(35,36,46,47)(40,42,44,41);(1,5)(2,9)(3,13)(4,10)(6,15)(7,16)(8,12)(11,14)(17,21,22,19)(18,23,25,26)(20,27,28,29)(24,32,31,30)(33,38)(34,37)(35,42)(36,40)(39,48)(41,46)(43,45)(44,47)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=h,d^2=gh,e^2=h,f^2=gh,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, 8641160);