Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1189 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(6,7)(9,10,14,19)(11,21,22,32)(12,26,28,34)(13,20,15,18)(16,25,30,24)(17,29,33,27)(23,37,36,31)(35,39,40,38);(1,2,3,4)(5,6)(7,8)(11,22)(16,30)(21,32)(23,36)(24,25)(31,37)(35,40)(38,39);(2,4)(5,6,8,7)(9,11,15,25,14,22,13,24)(10,16,20,32,19,30,18,21)(12,23,29,39,28,36,27,38)(17,31,34,40,33,37,26,35);(5,7,8,6)(9,12)(10,17)(11,23)(13,27)(14,28)(15,29)(16,31)(18,26)(19,33)(20,34)(21,35)(22,36)(24,38)(25,39)(30,37)(32,40)Polycyclic presentation:
2^9|a^2=i,b^2=egi,c^2=fi,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 345521);