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 | |
4153 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(18,22)(23,25)(24,28)(30,31)(33,34,39,41)(35,40,45,48)(36,42,43,37)(38,44,47,46);(2,7)(3,11)(4,14)(13,15)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32)(33,35,39,45)(34,40,41,48)(36,44,43,46)(37,38,42,47);(2,7)(4,14)(8,10)(12,16)(19,26)(23,25)(27,32)(30,31)(34,41)(35,45)(37,42)(44,46);(1,3,6,11)(2,8,7,10)(4,12,14,16)(5,15,9,13)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,31)(30,32)(33,36,39,43)(34,42,41,37)(35,46,45,44)(38,48,47,40);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,15)(11,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32)(33,37,39,42)(34,43,41,36)(35,47,45,38)(40,44,48,46);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,16,10,12)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,38,39,47)(34,44,41,46)(35,42,45,37)(36,40,43,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8529419);