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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4011 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(2,6)(7,8)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26,30,34)(27,35,36,31)(28,32,41,39)(29,42,43,33)(37,47,49,44)(38,45,50,48)(40,52,53,46)(51,54,56,55);(1,2)(3,7)(4,8)(5,6)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(27,36)(31,35)(37,49)(38,50)(44,47)(45,48)(51,56)(54,55);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,26)(27,31)(28,39)(29,33)(30,34)(32,41)(35,36)(37,47)(38,45)(40,52)(42,43)(44,49)(46,53)(48,50)(51,55)(54,56);(1,3,5,4)(2,7,6,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27)(26,31)(28,37)(29,38)(30,36)(32,44)(33,45)(34,35)(39,47)(40,51)(41,49)(42,48)(43,50)(46,54)(52,55)(53,56);(1,4,5,3)(2,8,6,7)(25,28)(26,32)(27,37)(29,40)(30,41)(31,44)(33,46)(34,39)(35,47)(36,49)(38,51)(42,52)(43,53)(45,54)(48,55)(50,56);(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,29,30,43)(26,33,34,42)(27,38,36,50)(28,40,41,53)(31,45,35,48)(32,46,39,52)(37,51,49,56)(44,54,47,55)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7551259);