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 | #58802 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4096 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)(3,7)(4,5)(6,8)(10,12)(11,16)(14,22)(21,24)(25,26,31,29)(27,35,36,32)(28,39,41,33)(30,42,44,34)(37,45,50,49)(38,46,43,48)(40,47,54,52)(51,56,53,55);(2,5)(6,7)(9,10,15,12)(11,18,16,13)(14,17,22,20)(19,24,23,21)(27,36)(32,35)(37,50)(38,43)(45,49)(46,48)(51,53)(55,56);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,18)(14,19)(15,16)(17,21)(20,24)(22,23)(25,27)(26,32)(28,37)(29,35)(30,43)(31,36)(33,45)(34,48)(38,44)(39,49)(40,53)(41,50)(42,46)(47,56)(51,54)(52,55);(1,3)(2,6)(4,8)(5,7)(9,12,15,10)(11,18,16,13)(14,20,22,17)(19,24,23,21)(25,28,31,41)(26,33,29,39)(27,37,36,50)(30,40,44,54)(32,45,35,49)(34,47,42,52)(38,51,43,53)(46,55,48,56);(9,13,15,18)(10,16,12,11)(14,21,22,24)(17,23,20,19)(25,29,31,26)(27,35,36,32)(28,33,41,39)(30,42,44,34)(37,45,50,49)(38,48,43,46)(40,47,54,52)(51,55,53,56);(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,30)(26,34)(27,38)(28,40)(29,42)(31,44)(32,46)(33,47)(35,48)(36,43)(37,51)(39,52)(41,54)(45,55)(49,56)(50,53)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833806);