Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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 | |
1880 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14,19,24,20,25,18,23)(15,26,17,34,28,40,35,21)(16,29,32,38,33,22,31,39)(27,41,30,36,42,37,44,43);(1,2)(3,4)(13,15,20,28)(14,21,25,34)(16,30,33,44)(17,19,35,18)(22,37,29,41)(23,26,24,40)(27,31,42,32)(36,38,43,39);(2,4)(6,9)(10,12)(13,16,19,32,20,33,18,31)(14,22,24,39,25,29,23,38)(15,27,17,30,28,42,35,44)(21,36,26,37,34,43,40,41);(2,4)(13,17,19,28,20,35,18,15)(14,21,24,26,25,34,23,40)(16,30,32,42,33,44,31,27)(22,36,39,37,29,43,38,41);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,23,25,24)(15,17,28,35)(16,31,33,32)(21,26,34,40)(22,38,29,39)(27,30,42,44)(36,37,43,41)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,d^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6363869);