Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×#509 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1415 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,21)(13,15)(16,18)(17,19)(20,22)(23,24);(1,3)(2,6)(4,7)(5,8);(1,4,5,2)(3,7,8,6)(10,14)(11,18)(12,19)(15,22)(16,17)(20,24)(21,23);(3,8)(6,7);(10,15)(14,22)(16,23)(17,21)Polycyclic presentation:
2^9|a^2=f,c^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=df,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #41×#509, #72342465, #72342468, … |
| maximal normal subgroup of | #41×#509, #72342465, #72342468, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531007);