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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3999 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(2,4)(6,8)(9,10)(11,17)(12,14)(13,20)(15,18)(16,21)(19,22)(23,24)(25,26,31,34)(27,35,36,28)(29,41,42,32)(30,44,45,33)(37,39,49,47)(38,40,50,48)(43,46,54,53)(51,55,56,52);(3,7)(6,8)(10,12)(11,18)(16,20)(19,24)(27,36)(28,35)(29,42)(32,41)(38,50)(40,48)(43,54)(46,53);(1,2,5,4)(3,6,7,8)(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24)(25,27,31,36)(26,28,34,35)(29,37,42,49)(30,38,45,50)(32,39,41,47)(33,40,44,48)(43,51,54,56)(46,52,53,55);(1,3)(2,6)(4,8)(5,7)(9,12,14,10)(11,17,18,15)(13,20,21,16)(19,23,24,22)(25,28)(26,27)(29,39)(30,40)(31,35)(32,37)(33,38)(34,36)(41,49)(42,47)(43,52)(44,50)(45,48)(46,51)(53,56)(54,55);(1,4,5,2)(3,8,7,6)(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,29,31,42)(26,32,34,41)(27,37,36,49)(28,39,35,47)(30,43,45,54)(33,46,44,53)(38,51,50,56)(40,52,48,55);(25,30,31,45)(26,33,34,44)(27,38,36,50)(28,40,35,48)(29,43,42,54)(32,46,41,53)(37,51,49,56)(39,52,47,55)Polycyclic presentation:
2^9|a^2=gi,c^2=gh,d^2=g,e^2=h,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, 8022660);