Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418500 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1425 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2,3,4)(6,10)(11,12)(13,14,20,26)(15,27,30,37)(16,31,35,40)(17,21,29,23)(18,25,19,24)(22,33,41,34)(28,36,43,39)(32,44,42,38);(5,6,9,10)(7,11,8,12)(13,15,20,30)(14,21,26,23)(16,32,35,42)(17,18,29,19)(22,38,41,44)(24,37,25,27)(28,34,43,33)(31,39,40,36);(2,4)(5,7,9,8)(6,11,10,12)(13,16,18,33,20,35,19,34)(14,22,24,40,26,41,25,31)(15,28,17,32,30,43,29,42)(21,36,27,38,23,39,37,44);(5,8,9,7)(6,11,10,12)(13,17,18,30,20,29,19,15)(14,23,24,37,26,21,25,27)(16,32,33,43,35,42,34,28)(22,39,40,44,41,36,31,38)Polycyclic presentation:
2^9|a^2=fgi,b^2=eg,c^2=e,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348321);