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 | #2703 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,22,23)(19,21,26,24)(20,28,30,25)(27,31,32,29)(33,34)(35,37)(36,44)(38,42)(39,40)(41,47)(43,46)(45,48);(1,2,6,7)(3,8,13,16)(4,9,14,12)(5,10,15,11)(18,23)(19,26)(25,28)(27,32)(34,40)(35,42)(41,44)(43,48);(2,7)(4,14)(8,16)(10,11)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,35,39,42)(34,37,40,38)(36,43,47,48)(41,45,44,46);(1,3)(2,8)(4,11)(5,9)(6,13)(7,16)(10,14)(12,15)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29)(33,36,39,47)(34,41,40,44)(35,43,42,48)(37,45,38,46);(1,4)(2,9)(3,11)(5,8)(6,14)(7,12)(10,13)(15,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,37,39,38)(34,35,40,42)(36,45,47,46)(41,43,44,48);(1,5)(2,10)(3,12)(4,8)(6,15)(7,11)(9,13)(14,16)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)(33,38,39,37)(34,42,40,35)(36,46,47,45)(41,48,44,43)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8424787);