Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501172 |
| 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 |
| prime radical | #25739 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1056 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)(14,21)(16,20)(18,23)(22,24);(2,5)(6,7)(9,10,13,15)(11,14,18,22)(12,20,19,16)(17,21,23,24);(9,11,12,17,13,18,19,23)(10,14,16,21,15,22,20,24);(1,3,4,8)(2,6,5,7)(10,15)(14,22)(16,20)(21,24)Polycyclic presentation:
2^9|b^2=i,c^2=f,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^c=gh,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72285710, #72285863, #72286069, … |
| maximal normal subgroup of | #72285710, #72285863, #72286069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410028);