Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
1231 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,30,35)(27,31,37,33)(28,39,42,55)(29,43,44,34)(32,47,49,41)(36,51,40,54)(38,50,53,46)(45,56,48,52);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,30,37)(26,31,35,33)(28,40,42,36)(29,38,44,53)(32,48,49,45)(34,46,43,50)(39,54,55,51)(41,56,47,52);(11,17)(14,21)(18,23)(22,24)(25,28,29,41,30,42,44,47)(26,32,34,39,35,49,43,55)(27,36,38,52,37,40,53,56)(31,45,46,51,33,48,50,54);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,27)(26,33)(28,40)(29,38)(30,37)(31,35)(32,45)(34,50)(36,42)(39,51)(41,56)(43,46)(44,53)(47,52)(48,49)(54,55);(1,3,4,7)(2,6,5,8)(27,37)(31,33)(36,40)(38,53)(45,48)(46,50)(51,54)(52,56)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343657);