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 | #44266 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4025 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)(3,7)(4,5)(6,8)(9,13)(10,11)(12,14)(15,16)(17,18,23,25)(19,24,27,21)(20,30,26,22)(28,29,31,32)(33,34)(35,44)(36,38)(37,42)(39,40)(41,47)(43,48)(45,46);(1,3,6,12)(2,7,8,14)(4,10,9,15)(5,11,13,16)(17,19,23,27)(18,24,25,21)(20,31,26,28)(22,29,30,32)(34,40)(36,43)(41,44)(42,45);(2,8)(4,9)(7,14)(10,15)(18,25)(19,27)(20,26)(29,32)(33,35,39,47)(34,41,40,44)(36,42,43,45)(37,38,46,48);(1,4,6,9)(2,5,8,13)(3,10,12,15)(7,11,14,16)(17,20,23,26)(18,22,25,30)(19,28,27,31)(21,29,24,32)(33,36,39,43)(34,38,40,48)(35,45,47,42)(37,44,46,41);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,21)(18,19)(20,29)(22,28)(23,24)(25,27)(26,32)(30,31)(33,37)(34,42)(35,38)(36,44)(39,46)(40,45)(41,43)(47,48);(1,5)(2,9)(3,11)(4,8)(6,13)(7,15)(10,14)(12,16)(17,22)(18,26)(19,29)(20,25)(21,31)(23,30)(24,28)(27,32)(33,38)(34,43)(35,46)(36,40)(37,47)(39,48)(41,45)(42,44)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9843659);