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 | |
4102 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)(3,4)(5,9)(6,7)(8,11)(10,12)(13,16)(14,15)(17,18,23,24)(19,20,21,25)(22,30,31,26)(27,32,29,28)(34,40)(35,44)(36,42)(38,47);(2,7)(4,8)(9,15)(12,13)(18,24)(19,21)(26,30)(27,29)(33,34,39,40)(35,41,44,48)(36,37,42,45)(38,43,47,46);(17,19,23,21)(18,25,24,20)(22,27,31,29)(26,32,30,28)(33,35,39,44)(34,41,40,48)(36,46,42,43)(37,38,45,47);(1,3)(2,8)(4,7)(5,10)(6,11)(9,13)(12,15)(14,16)(17,20,23,25)(18,19,24,21)(22,28,31,32)(26,27,30,29)(33,36,39,42)(34,37,40,45)(35,43,44,46)(38,41,47,48);(1,4,6,8)(2,3,7,11)(5,13,14,12)(9,16,15,10)(17,21,23,19)(18,25,24,20)(22,29,31,27)(26,32,30,28)(33,37)(34,42)(35,38)(36,40)(39,45)(41,43)(44,47)(46,48);(1,5,6,14)(2,9,7,15)(3,10,11,16)(4,12,8,13)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(33,38)(34,43)(35,45)(36,41)(37,44)(39,47)(40,46)(42,48)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=h,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9754819);