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 | #45952 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4093 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)(9,10,15,12)(11,18,17,14)(13,16,21,20)(19,24,23,22)(27,35)(31,36)(37,40)(38,49)(45,47)(46,50)(51,53)(55,56);(2,5)(6,7)(13,21)(16,20)(19,23)(22,24)(25,26,30,34)(27,31,35,36)(28,39,42,32)(29,33,44,43)(37,47,40,45)(38,46,49,50)(41,52,54,48)(51,56,53,55);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,24)(21,23)(25,26)(27,36)(28,32)(29,43)(30,34)(31,35)(33,44)(37,47)(38,46)(39,42)(40,45)(41,52)(48,54)(49,50)(51,55)(53,56);(1,3)(2,6)(4,8)(5,7)(9,12,15,10)(11,18,17,14)(13,20,21,16)(19,24,23,22)(25,27,30,35)(26,31,34,36)(28,40,42,37)(29,38,44,49)(32,47,39,45)(33,46,43,50)(41,53,54,51)(48,56,52,55);(9,13)(10,16)(11,19)(12,20)(14,22)(15,21)(17,23)(18,24)(25,28,30,42)(26,32,34,39)(27,37,35,40)(29,41,44,54)(31,45,36,47)(33,48,43,52)(38,51,49,53)(46,55,50,56);(9,14,15,18)(10,17,12,11)(13,22,21,24)(16,23,20,19)(25,29)(26,33)(27,38)(28,41)(30,44)(31,46)(32,48)(34,43)(35,49)(36,50)(37,51)(39,52)(40,53)(42,54)(45,55)(47,56)Polycyclic presentation:
2^9|a^2=h,b^2=gi,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839746);