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 | |
4299 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)(3,4)(5,13)(6,7)(8,9)(10,14)(11,15)(12,16)(17,18)(19,25)(20,28)(21,22)(23,26)(24,30)(27,31)(29,32)(36,43)(39,41)(44,48)(46,47);(2,7)(4,8)(5,14)(11,16)(18,22)(23,25)(24,28)(31,32)(33,34,38,35)(36,39,43,41)(37,42,45,40)(44,47,48,46);(2,7)(3,9)(10,13)(11,16)(19,26)(23,25)(27,29)(31,32)(34,35)(39,41)(40,42)(46,47);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,13)(14,16)(17,19,21,26)(18,23,22,25)(20,29,30,27)(24,32,28,31)(33,35,38,34)(36,39,43,41)(37,42,45,40)(44,46,48,47);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,20,21,30)(18,24,22,28)(19,27,26,29)(23,31,25,32)(33,36)(34,39)(35,41)(37,44)(38,43)(40,46)(42,47)(45,48);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|b^2=i,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8748901);