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 | #42674 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(2,5)(3,4)(9,10,14,15)(11,12,17,20)(13,16,21,22)(18,19,23,24)(25,26)(27,33)(28,29)(30,44)(31,32)(34,42)(35,45)(36,48)(37,40)(38,52)(39,46)(41,49)(43,55)(47,53)(50,51)(54,56);(1,2,6,5)(3,8,4,7)(10,15)(11,17)(16,22)(18,23)(26,32)(29,34)(33,46)(35,44)(37,48)(43,49)(47,52)(51,56);(1,3,6,4)(2,7,5,8)(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24)(25,27)(26,33)(28,40)(29,37)(30,38)(31,39)(32,46)(34,48)(35,47)(36,42)(41,54)(43,51)(44,52)(45,53)(49,56)(50,55);(1,4,6,3)(2,8,5,7)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,28)(26,34)(27,36)(29,32)(30,41)(31,42)(33,37)(35,49)(38,50)(39,40)(43,44)(45,55)(46,48)(47,51)(52,56)(53,54);(1,5,6,2)(3,8,4,7)(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,19,20,24)(25,29,31,34)(26,28,32,42)(27,37,39,48)(30,43,45,49)(33,40,46,36)(35,41,44,55)(38,51,53,56)(47,54,52,50);(25,30,31,45)(26,35,32,44)(27,38,39,53)(28,41,42,55)(29,43,34,49)(33,47,46,52)(36,50,40,54)(37,51,48,56)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=h,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, 8022624);