Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(13,14,20,26)(15,22,29,36)(16,21,28,37)(17,23,32,40)(18,25,19,24)(27,38,43,39)(30,35,31,44)(33,42,34,41);(5,6)(7,10)(8,9)(11,12)(13,15,19,28,20,29,18,16)(14,21,25,36,26,37,24,22)(17,27,34,31,32,43,33,30)(23,35,42,39,40,44,41,38);(2,4)(6,9)(7,11)(13,16,18,29,20,28,19,15)(14,22,24,37,26,36,25,21)(17,31,33,27,32,30,34,43)(23,39,41,35,40,38,42,44);(5,7,8,11)(6,10,9,12)(13,17,20,32)(14,23,26,40)(15,27,29,43)(16,30,28,31)(18,33,19,34)(21,35,37,44)(22,38,36,39)(24,41,25,42);(6,9)(10,12)(17,32)(23,40)(27,43)(30,31)(33,34)(35,44)(38,39)(41,42)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6421718);