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 | #47932 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1203 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,19)(14,16)(15,22)(17,23)(18,20)(21,24);(1,3,5,4)(2,6,8,7)(9,11,13,10)(12,17,21,15)(14,16,20,18)(19,22,24,23);(1,3,5,4)(2,7,8,6)(9,12,20,23)(10,15,18,24)(11,17,16,19)(13,21,14,22);(1,4,5,3)(2,7,8,6)(10,11)(15,17)(16,18)(19,24)Polycyclic presentation:
2^9|b^2=e,c^2=ef,d^2=ei,b^a=be,c^a=cf,d^a=de,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 | #72251961, #72251968, #72251974, … |
| maximal normal subgroup of | #72251961, #72251968, #72251974, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410681);