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 | |
4133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,11,9)(4,12,13,8)(10,15,16,14)(17,18)(19,22)(20,27)(21,24)(23,28)(25,31)(26,30)(29,32)(33,34,40,44)(35,45,48,41)(36,42,37,47)(38,43,46,39);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31)(33,35)(34,41)(36,46)(37,38)(39,47)(40,48)(42,43)(44,45);(2,6)(7,9)(8,12)(14,15)(17,18,21,24)(19,22,26,30)(20,23,28,27)(25,29,32,31)(33,36,40,37)(34,42,44,47)(35,46,48,38)(39,41,43,45);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,37,40,36)(34,42,44,47)(35,46,48,38)(39,45,43,41);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,38,40,46)(34,43,44,39)(35,37,48,36)(41,42,45,47);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,20,21,28)(18,23,24,27)(19,25,26,32)(22,29,30,31)(33,39,40,43)(34,38,44,46)(35,47,48,42)(36,41,37,45)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=gh,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154804);