Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470395 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1308 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)(3,7)(4,5)(6,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,36)(28,33)(29,43)(30,35)(31,39)(32,41)(34,37)(38,50)(40,51)(42,47)(44,56)(45,55)(46,52)(48,53)(49,54);(2,5)(6,7)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,37,52,30,40,29,39)(26,31,43,51,35,46,34,36)(28,38,53,56,41,54,42,55)(32,44,48,50,33,45,47,49);(1,3,4,8)(2,6,5,7)(11,17)(14,21)(18,23)(22,24)(25,28)(26,32)(27,38)(29,42)(30,41)(31,44)(33,35)(34,47)(36,49)(37,53)(39,55)(40,54)(43,48)(45,46)(50,51)(52,56);(11,17)(14,21)(18,23)(22,24)(25,28)(26,33)(27,38)(29,42)(30,41)(31,45)(32,35)(34,48)(36,50)(37,53)(39,55)(40,54)(43,47)(44,46)(49,51)(52,56);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(28,41)(32,33)(38,54)(42,53)(44,45)(47,48)(49,50)(55,56)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=gi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405218);