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 | |
4324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,5)(3,9,11,7)(4,8,12,13)(10,15,16,14)(17,18)(19,21)(20,23)(22,28)(24,26)(25,30)(27,31)(29,32)(33,34)(35,39)(36,41)(37,42)(38,43)(40,45)(44,47)(46,48);(1,3,6,11)(2,7,5,9)(4,10,12,16)(8,14,13,15)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32)(33,35)(34,39)(36,42)(37,41)(38,45)(40,43)(44,46)(47,48);(4,12)(8,13)(10,16)(14,15)(18,20)(19,26)(25,28)(27,32)(33,35,38,45)(34,39,43,40)(36,44,46,42)(37,41,47,48);(1,4,6,12)(2,8,5,13)(3,10,11,16)(7,14,9,15)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,35)(34,40)(36,44)(37,41)(38,45)(39,43)(42,46)(47,48);(1,3)(2,9)(4,10)(5,7)(6,11)(8,15)(12,16)(13,14)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,36)(34,41)(35,44)(37,40)(38,46)(39,47)(42,45)(43,48);(1,5,6,2)(3,9,11,7)(4,13,12,8)(10,15,16,14)(17,22,23,30)(18,25,20,28)(19,27,26,32)(21,29,24,31)(33,37)(34,42)(35,41)(36,39)(38,47)(40,46)(43,44)(45,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=hi,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dghi,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9877311);