Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ #1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,3,4,6,8,7)(9,10,15,21)(11,23,12,18)(13,22,16,19)(14,29,31,37)(17,27,34,25)(20,28,38,32)(24,35,26,36)(30,39,40,33);(2,6)(3,7)(9,11,15,12)(10,17,21,34)(13,24,16,26)(14,30,31,40)(18,29,23,37)(19,33,22,39)(20,36,38,35)(25,28,27,32);(1,3,5,6,4,7,8,2)(9,12,15,11)(10,18,21,23)(13,24,16,26)(14,27,31,25)(17,29,34,37)(19,36,22,35)(20,33,38,39)(28,30,32,40);(9,13,15,16)(10,19,21,22)(11,24,12,26)(14,28,31,32)(17,33,34,39)(18,35,23,36)(20,37,38,29)(25,30,27,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=di,e^a=eg,g^a=gi,c^b=ch,e^b=eg,g^b=gi,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72232015, #72232022, #72232025, … |
| maximal normal subgroup of | #72232015, #72232022, #72232025, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391028);