Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #491712 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×D8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,8)(6,7)(9,10,15,19)(11,20,23,38)(12,17,14,29)(13,25,28,39)(16,31,34,22)(18,24,36,27)(21,37,30,35)(26,32,40,33);(1,2)(3,6)(4,7)(5,8);(9,11)(10,16)(12,21)(13,26)(14,30)(15,23)(17,32)(18,35)(19,34)(20,25)(22,27)(24,31)(28,40)(29,33)(36,37)(38,39);(2,6)(3,8)(5,7)(9,12,15,14)(10,17,19,29)(11,21,23,30)(13,24,28,27)(16,32,34,33)(18,25,36,39)(20,37,38,35)(22,26,31,40)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,d^b=de,e^b=eg,f^c=fh,h^c=hi,e^d=eg62 conjugacy classes — view the full character table.
| maximal subgroup of | #72310003 |
| maximal normal subgroup of | #72310003 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 398917);