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 | |
4158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(5,13)(7,9)(11,14)(20,28)(23,29)(26,32)(30,31)(33,34,39,44)(35,45,47,40)(36,42,43,38)(37,41,46,48);(2,3)(7,9)(8,10)(15,16)(17,18)(19,24)(20,29)(21,25)(22,27)(23,28)(26,30)(31,32)(33,35,39,47)(34,40,44,45)(36,41,43,48)(37,38,46,42);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,15,14,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,32)(29,31)(33,36)(34,38)(35,41)(37,45)(39,43)(40,46)(42,44)(47,48);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,15,14,16)(17,20,21,28)(18,23,25,29)(19,26,27,32)(22,30,24,31)(33,37,39,46)(34,41,44,48)(35,42,47,38)(36,45,43,40);(1,4)(2,7)(3,9)(5,14)(6,12)(8,16)(10,15)(11,13)(17,19)(18,24)(20,26)(21,27)(22,25)(23,31)(28,32)(29,30)(33,36)(34,42)(35,41)(37,40)(38,44)(39,43)(45,46)(47,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,15)(9,16)(12,14)(17,18)(19,22)(20,23)(21,25)(24,27)(26,30)(28,29)(31,32)(33,38,39,42)(34,43,44,36)(35,46,47,37)(40,48,45,41)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10388110);