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 | #2678 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,9)(3,7,10,6)(4,12,13,8)(11,14,16,15)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32)(37,45)(42,46)(43,44)(47,48);(3,10)(6,7)(11,16)(14,15)(18,23)(19,26)(25,28)(27,32)(34,39)(40,41)(42,46)(47,48);(1,3,5,10)(2,6,9,7)(4,11,13,16)(8,14,12,15)(17,18,22,23)(19,24,26,21)(20,25,30,28)(27,31,32,29)(33,34)(35,41)(36,40)(37,42)(38,39)(43,48)(44,47)(45,46);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,35,38,36)(34,40,39,41)(37,43,45,44)(42,47,46,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,15)(12,13)(14,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,36,38,35)(34,41,39,40)(37,44,45,43)(42,48,46,47);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)(33,37)(34,42)(35,43)(36,44)(38,45)(39,46)(40,47)(41,48)Polycyclic presentation:
2^9|a^2=gh,c^2=h,d^2=hi,e^2=ghi,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^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8038385);