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 | |
4063 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,8,6,10)(5,9,14,12)(11,15,13,16)(17,18)(19,23)(20,27)(21,26)(22,32)(24,29)(25,31)(28,30)(33,34,39,38)(35,41,45,40)(36,46,44,42)(37,43,48,47);(1,3,7,6)(2,8,4,10)(5,11,14,13)(9,15,12,16)(17,19)(18,25)(20,30)(21,22)(23,29)(24,31)(26,28)(27,32)(33,35,39,45)(34,40,38,41)(36,37,44,48)(42,43,46,47);(3,6)(8,10)(11,13)(15,16)(17,20,24,21)(18,26,29,27)(19,30,31,22)(23,32,25,28)(33,35,39,45)(34,41,38,40)(36,37,44,48)(42,47,46,43);(1,4,7,2)(3,10,6,8)(5,9,14,12)(11,15,13,16)(17,21,24,20)(18,27,29,26)(19,30,31,22)(23,32,25,28)(33,36)(34,42)(35,37)(38,46)(39,44)(40,43)(41,47)(45,48);(1,5,7,14)(2,9,4,12)(3,11,6,13)(8,15,10,16)(17,22,24,30)(18,28,29,32)(19,21,31,20)(23,27,25,26)(33,37)(34,43)(35,44)(36,45)(38,47)(39,48)(40,46)(41,42);(1,6,7,3)(2,10,4,8)(5,13,14,11)(9,16,12,15)(17,23,24,25)(18,19,29,31)(20,28,21,32)(22,27,30,26)(33,38,39,34)(35,41,45,40)(36,46,44,42)(37,47,48,43)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=hi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8253116);