Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628867 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23);(1,3,4,8)(2,5,6,7);(9,11)(10,14)(12,21)(13,19)(15,23)(16,24)(17,20)(18,22);(12,22)(16,20)(17,24)(18,21);(10,15)(14,23)(16,20)(17,24)Polycyclic presentation:
2^9|a^2=f,b^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,g^c=gi,h^c=hi,g^d=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239476, #72257517, #72257542, … |
| maximal normal subgroup of | #72239476, #72257517, #72257542, … |
| maximal quotient of | #72257958, #72257967 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536304);