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 | |
4169 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,28)(20,21)(22,32)(23,25)(24,29)(26,31)(27,30)(33,34)(35,41)(36,40)(37,42)(38,39)(43,45)(44,46)(47,48);(3,9)(7,13)(10,15)(14,16)(17,19,23,29)(18,24,25,28)(20,30,31,22)(21,32,26,27)(33,34)(35,37)(36,44)(38,39)(40,46)(41,42)(43,45)(47,48);(2,6)(7,13)(8,11)(14,16)(18,25)(19,29)(20,31)(27,32)(34,39)(35,42)(40,44)(43,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20)(18,26)(19,30)(21,25)(22,29)(23,31)(24,27)(28,32)(33,35)(34,37)(36,43)(38,42)(39,41)(40,47)(44,45)(46,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,21,23,26)(18,20,25,31)(19,27,29,32)(22,28,30,24)(33,36)(34,40)(35,43)(37,47)(38,46)(39,44)(41,45)(42,48);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,22,23,30)(18,27,25,32)(19,20,29,31)(21,28,26,24)(33,37)(34,35)(36,45)(38,41)(39,42)(40,48)(43,44)(46,47)Polycyclic presentation:
2^9|b^2=g,e^2=g,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, 9621823);