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 | |
4112 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,23)(20,24)(21,30)(22,25)(26,28)(27,31)(29,32)(33,34)(35,45)(36,42)(37,38)(39,43)(40,44)(41,48)(46,47);(3,9)(7,13)(10,15)(14,16)(17,19,22,29)(18,23,25,32)(20,27,30,26)(21,28,24,31)(33,35,40,48)(34,41,44,45)(36,46,37,39)(38,47,42,43);(2,6)(7,13)(8,11)(14,16)(17,20,22,30)(18,21,25,24)(19,26,29,27)(23,31,32,28)(33,36,40,37)(34,42,44,38)(35,46,48,39)(41,43,45,47);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20)(18,24)(19,27)(21,25)(22,30)(23,31)(26,29)(28,32)(33,37,40,36)(34,42,44,38)(35,39,48,46)(41,43,45,47);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,19)(18,23)(20,26)(21,31)(22,29)(24,28)(25,32)(27,30)(33,38,40,42)(34,37,44,36)(35,47,48,43)(39,45,46,41);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,24)(18,20,25,30)(19,28,29,31)(23,27,32,26)(33,39)(34,43)(35,36)(37,48)(38,41)(40,46)(42,45)(44,47)Polycyclic presentation:
2^9|b^2=gi,c^2=gi,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10103175);