Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508066 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1063 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(10,12)(11,17)(15,22)(16,20)(18,24)(19,23);(2,5)(6,8)(9,10,13,12)(11,15,18,19)(14,20,21,16)(17,23,24,22);(3,7)(6,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,18,19)(14,20,21,16)(17,23,24,22)Polycyclic presentation:
2^9|b^2=i,d^2=e,b^a=be,c^a=cf,d^a=di,g^a=ghi,c^b=cg,f^b=fh,d^c=de,h^c=hi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72278373, #72278403, #72283327, … |
| maximal normal subgroup of | #72278373, #72278403, #72283327, … |
| maximal quotient of | #72278791 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415402);