Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433044 |
| 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 | |
1273 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)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24)(25,26,30,33)(27,31,36,45)(28,38,42,55)(29,43,40,34)(32,47,49,41)(35,50,39,54)(37,53,51,46)(44,56,48,52);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(25,27)(26,31)(28,39)(29,37)(30,36)(32,48)(33,45)(34,46)(35,42)(38,54)(40,51)(41,56)(43,53)(44,49)(47,52)(50,55);(11,17)(14,20)(18,23)(21,24)(25,28,40,47,30,42,29,41)(26,32,43,55,33,49,34,38)(27,35,51,56,36,39,37,52)(31,44,53,54,45,48,46,50);(1,3,4,7)(2,6,5,8)(10,15)(14,20)(16,22)(21,24)(26,33)(31,45)(32,49)(34,43)(38,55)(44,48)(46,53)(50,54);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)(27,36)(31,45)(35,39)(37,51)(44,48)(46,53)(50,54)(52,56)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,e^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344523);