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 | |
4246 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,30)(31,32)(35,42)(37,39)(43,48)(45,47);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19,23,30)(18,24,28,29)(20,21,27,31)(22,26,25,32)(33,34)(35,37)(36,44)(38,41)(39,42)(40,46)(43,45)(47,48);(2,7)(6,13)(8,11)(14,15)(33,35,38,42)(34,39,41,37)(36,43,46,48)(40,45,44,47);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20,23,27)(18,25,28,22)(19,21,30,31)(24,32,29,26)(33,35)(34,37)(36,43)(38,42)(39,41)(40,47)(44,45)(46,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,21)(18,26)(19,27)(20,30)(22,29)(23,31)(24,25)(28,32)(33,36,38,46)(34,40,41,44)(35,43,42,48)(37,47,39,45);(1,4,5,12)(2,8,7,11)(3,9,10,16)(6,14,13,15)(17,22,23,25)(18,27,28,20)(19,26,30,32)(21,24,31,29)(33,37)(34,35)(36,45)(38,39)(40,48)(41,42)(43,44)(46,47)Polycyclic presentation:
2^9|b^2=ghi,c^2=i,d^2=g,e^2=i,f^2=ghi,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, 9636395);