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 |
| prime radical | #56376 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4188 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(10,16)(14,15)(17,18,23,21)(19,26,22,24)(20,28,29,25)(27,31,30,32)(33,34,38,42)(35,40,36,39)(37,45,46,41)(43,47,44,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19,23,22)(18,24,21,26)(20,27,29,30)(25,31,28,32)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(2,6)(7,9)(8,13)(14,15)(20,29)(25,28)(27,30)(31,32)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20,23,29)(18,25,21,28)(19,27,22,30)(24,31,26,32)(33,34)(35,40)(36,39)(37,41)(38,42)(43,48)(44,47)(45,46);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,21,23,18)(19,26,22,24)(20,25,29,28)(27,31,30,32)(33,36,38,35)(34,40,42,39)(37,44,46,43)(41,48,45,47);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,22,23,19)(18,26,21,24)(20,30,29,27)(25,32,28,31)(33,37,38,46)(34,41,42,45)(35,43,36,44)(39,47,40,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8062252);