Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433057 |
| 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 | |
1369 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)(25,26,31,36)(27,37,40,33)(28,32,42,47)(29,34,43,49)(30,45,38,35)(39,51,53,48)(41,46,54,56)(44,55,52,50);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(25,27,38,42,31,40,30,28)(26,32,45,33,36,47,35,37)(29,39,52,54,43,53,44,41)(34,46,55,48,49,56,50,51);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)(25,28,30,40,31,42,38,27)(26,33,35,32,36,37,45,47)(29,41,44,53,43,54,52,39)(34,48,50,46,49,51,55,56);(10,15)(14,20)(16,22)(21,24)(25,29,31,43)(26,34,36,49)(27,39,40,53)(28,41,42,54)(30,44,38,52)(32,46,47,56)(33,48,37,51)(35,50,45,55);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)(29,43)(34,49)(39,53)(41,54)(44,52)(46,56)(48,51)(50,55)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=f,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344396);