Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,4)(5,6,7,8)(10,16)(12,24)(13,26)(14,28)(17,33)(18,37)(19,39)(20,34)(21,31)(22,38)(25,35)(27,30)(29,32)(36,40);(9,10,15,20)(11,17,23,36)(12,25,21,37)(13,27,29,39)(14,30,28,19)(16,32,34,26)(18,38,35,22)(24,40,31,33);(1,2,3,4)(9,11,15,23)(10,17,20,36)(12,13,21,29)(14,22,28,38)(16,33,34,40)(18,19,35,30)(24,26,31,32)(25,27,37,39);(9,12,15,21)(10,18,20,35)(11,13,23,29)(14,31,28,24)(16,33,34,40)(17,19,36,30)(22,32,38,26)(25,39,37,27)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fi,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405071);