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 | #56272 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4097 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,8,10,6)(5,9,15,12)(11,16,13,14)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32)(33,34,39,37)(35,36,43,40)(38,41,46,45)(42,44,48,47);(1,3,7,10)(2,8,4,6)(5,13,15,11)(9,14,12,16)(18,23)(19,26)(25,28)(27,32)(33,35,39,43)(34,40,37,36)(38,42,46,48)(41,47,45,44);(3,10)(6,8)(11,13)(14,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,34,39,37)(35,40,43,36)(38,45,46,41)(42,44,48,47);(1,4,7,2)(3,6,10,8)(5,9,15,12)(11,16,13,14)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,36)(34,35)(37,43)(38,44)(39,40)(41,42)(45,48)(46,47);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,16)(10,13)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,37,39,34)(35,40,43,36)(38,45,46,41)(42,47,48,44);(1,6)(2,3)(4,10)(5,14)(7,8)(9,11)(12,13)(15,16)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)(33,38,39,46)(34,41,37,45)(35,42,43,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=gh,d^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8267171);