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 |
| prime radical | #40162 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(9,14)(10,15)(11,16)(17,18,24,23)(19,28,22,25)(20,27,21,26)(29,31,30,32)(33,34,39,42)(35,37,41,38)(36,44,47,40)(43,46,48,45);(1,2)(3,4)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(17,19,24,22)(18,25,23,28)(20,30,21,29)(26,32,27,31)(33,35,39,41)(34,38,42,37)(36,43,47,48)(40,46,44,45);(2,7)(3,8)(9,14)(10,15)(17,20,24,21)(18,26,23,27)(19,29,22,30)(25,31,28,32)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,46)(47,48);(1,3,6,8)(2,4,7,12)(5,10,13,15)(9,11,14,16)(17,21,24,20)(18,27,23,26)(19,30,22,29)(25,32,28,31)(33,36,39,47)(34,40,42,44)(35,43,41,48)(37,45,38,46);(1,4)(2,8)(3,7)(5,11)(6,12)(9,15)(10,14)(13,16)(17,22,24,19)(18,28,23,25)(20,30,21,29)(26,32,27,31)(33,37,39,38)(34,41,42,35)(36,45,47,46)(40,48,44,43);(1,5)(2,9)(3,10)(4,11)(6,13)(7,14)(8,15)(12,16)(17,23,24,18)(19,28,22,25)(20,27,21,26)(29,32,30,31)(33,38,39,37)(34,35,42,41)(36,46,47,45)(40,43,44,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8019764);