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 | #2812 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 56 points:
(3,7)(5,8)(9,10)(11,13)(12,20)(14,18)(15,17)(16,22)(19,23)(21,24)(26,31)(27,35)(29,42)(33,38)(36,39)(40,54)(46,51)(55,56);(1,2,4,6)(3,5,7,8)(9,10,14,18)(11,13,17,15)(12,16,22,20)(19,21,24,23)(26,31)(32,44)(33,38)(34,43)(45,47)(46,51)(48,52)(55,56);(1,3)(2,5)(4,7)(6,8)(9,11,14,17)(10,15,18,13)(12,19,22,24)(16,23,20,21)(25,26)(27,32)(28,38)(29,43)(30,31)(33,41)(34,42)(35,44)(36,47)(37,51)(39,45)(40,48)(46,49)(50,55)(52,54)(53,56);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20)(25,27,30,35)(26,32,31,44)(28,39,41,36)(29,37,42,49)(33,47,38,45)(34,46,43,51)(40,53,54,50)(48,56,52,55);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,22)(21,24)(25,28,30,41)(26,33,31,38)(27,36,35,39)(29,40,42,54)(32,45,44,47)(34,48,43,52)(37,50,49,53)(46,55,51,56);(1,3)(2,5)(4,7)(6,8)(9,13,14,15)(10,17,18,11)(12,21,22,23)(16,24,20,19)(25,29)(26,34)(27,37)(28,40)(30,42)(31,43)(32,46)(33,48)(35,49)(36,50)(38,52)(39,53)(41,54)(44,51)(45,55)(47,56)Polycyclic presentation:
2^9|b^2=ghi,c^2=g,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947314);