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 | |
4089 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)(12,13,14,16)(17,18,23,27)(19,21,26,24)(20,29,22,25)(28,30,32,31)(33,34)(35,41)(36,47)(37,40)(38,42)(39,44)(43,46)(45,48);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,14)(15,16)(17,19,23,26)(18,24,27,21)(20,28,22,32)(25,30,29,31)(33,35,40,48)(34,41,37,45)(36,43,42,44)(38,39,47,46);(3,11)(7,8)(12,14)(13,16)(17,20,23,22)(18,25,27,29)(19,28,26,32)(21,31,24,30)(33,36,40,42)(34,38,37,47)(35,44,48,43)(39,41,46,45);(1,3)(2,8)(4,14)(5,13)(6,11)(7,10)(9,12)(15,16)(17,21,23,24)(18,26,27,19)(20,30,22,31)(25,28,29,32)(33,37,40,34)(35,45,48,41)(36,38,42,47)(39,44,46,43);(1,4)(2,5)(3,12)(6,9)(7,13)(8,16)(10,15)(11,14)(17,22,23,20)(18,25,27,29)(19,28,26,32)(21,30,24,31)(33,38)(34,42)(35,46)(36,37)(39,48)(40,47)(41,43)(44,45);(1,5)(2,9)(3,13)(4,10)(6,15)(7,14)(8,12)(11,16)(17,18)(19,24)(20,25)(21,26)(22,29)(23,27)(28,30)(31,32)(33,39,40,46)(34,43,37,44)(35,47,48,38)(36,41,42,45)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9080351);