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 | #45316 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4089 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,32)(27,28,38,33)(29,41,42,34)(30,44,45,35)(36,46,49,39)(37,47,51,40)(43,48,54,53)(50,52,56,55);(2,3)(6,7)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(26,32)(28,33)(29,42)(35,44)(36,49)(40,47)(43,54)(50,56);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,27,31,38)(26,33,32,28)(29,36,42,49)(30,37,45,51)(34,46,41,39)(35,47,44,40)(43,50,54,56)(48,55,53,52);(1,3,5,2)(4,7,8,6)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,28,31,33)(26,27,32,38)(29,39,42,46)(30,40,45,47)(34,36,41,49)(35,37,44,51)(43,52,54,55)(48,50,53,56);(1,4,5,8)(2,6,3,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,29,31,42)(26,34,32,41)(27,36,38,49)(28,39,33,46)(30,43,45,54)(35,48,44,53)(37,50,51,56)(40,52,47,55);(25,30,31,45)(26,35,32,44)(27,37,38,51)(28,40,33,47)(29,43,42,54)(34,48,41,53)(36,50,49,56)(39,52,46,55)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=h,d^2=g,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, 8022575);