Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499787 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #32920 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1068 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)(3,7)(4,5)(6,8)(11,17)(12,19)(14,22)(15,20)(18,21)(23,24);(2,5)(6,7)(9,10,13,16)(11,14,18,23)(12,20,19,15)(17,22,21,24);(9,11)(10,14)(12,21)(13,18)(15,24)(16,23)(17,19)(20,22);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^9|b^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,f^c=fi,h^c=hi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72311192, #72311234, #72311256, … |
| maximal normal subgroup of | #72311192, #72311234, #72311256, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407697);