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 | #2431 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,9,12)(5,13,14,8)(10,16,15,11)(21,29)(25,30)(26,31)(27,32)(33,34,37,38)(35,41,42,39)(36,44,45,40)(43,46,48,47);(1,3,6,9)(2,4,7,12)(5,10,14,15)(8,11,13,16)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,30)(31,32)(34,38)(35,42)(40,44)(43,48);(2,7)(3,9)(8,13)(10,15)(18,23)(19,24)(25,30)(26,31)(33,35,37,42)(34,39,38,41)(36,43,45,48)(40,46,44,47);(1,3)(2,4)(5,10)(6,9)(7,12)(8,11)(13,16)(14,15)(17,19,22,24)(18,20,23,28)(21,26,29,31)(25,27,30,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,4)(2,3)(5,11)(6,12)(7,9)(8,10)(13,15)(14,16)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32)(33,36,37,45)(34,40,38,44)(35,43,42,48)(39,46,41,47);(1,5,6,14)(2,8,7,13)(3,10,9,15)(4,11,12,16)(17,21,22,29)(18,25,23,30)(19,26,24,31)(20,27,28,32)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=hi,e^2=g,f^2=gi,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, 8022774);