Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6631116 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1385 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,31,46,30,36,29,35)(27,37,40,50,41,32,42,49)(28,34,43,33,44,51,45,52)(38,48,56,54,39,53,55,47);(1,3,4,8)(2,5,6,7)(10,14)(15,20)(16,21)(22,24)(25,27)(26,32)(28,38)(29,42)(30,41)(31,40)(33,47)(34,53)(35,50)(36,37)(39,44)(43,56)(45,55)(46,49)(48,51)(52,54);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,30,44)(26,33,36,52)(27,38,41,39)(29,43,31,45)(32,47,37,54)(34,35,51,46)(40,55,42,56)(48,49,53,50);(2,6)(5,7)(11,17)(15,20)(18,23)(22,24)(25,28)(26,34)(27,39)(29,45)(30,44)(31,43)(32,48)(33,46)(35,52)(36,51)(37,53)(38,41)(40,55)(42,56)(47,50)(49,54);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,30,31)(26,35,36,46)(27,40,41,42)(28,43,44,45)(32,49,37,50)(33,51,52,34)(38,55,39,56)(47,53,54,48)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,c^b=ch,d^b=di,e^b=ei,d^c=dfh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6539089);