Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #488889 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | D4×#135 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
941 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,15,20)(11,21,24,38)(12,19,14,17)(13,26,29,34)(16,31,33,23)(18,25,36,28)(22,39,30,35)(27,37,40,32);(2,6)(3,4)(7,8)(9,11)(10,16)(12,22)(13,27)(14,30)(15,24)(17,32)(18,35)(19,37)(20,33)(21,26)(23,28)(25,31)(29,40)(34,38)(36,39);(2,7)(3,4)(6,8);(9,12,15,14)(10,17,20,19)(11,22,24,30)(13,25,29,28)(16,32,33,37)(18,34,36,26)(21,35,38,39)(23,27,31,40)Polycyclic presentation:
2^9|a^2=hi,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72309443, #72309456, #72309460, … |
| maximal normal subgroup of | #72309443, #72309456, #72309460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396146);