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 | |
4317 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,18,20,24)(12,17,14,21)(16,22,23,19);(2,5)(6,8)(10,15)(16,23)(17,21)(18,24);(9,11)(10,16)(12,19)(13,20)(14,22)(15,23)(17,24)(18,21);(12,14)(17,21)(18,24)(19,22);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,17,15,21)(11,19,20,22)(16,24,23,18)Polycyclic presentation:
2^9|a^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,g^c=gi,h^c=hi,e^d=ei,g^d=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239476, #72278399, #72278497, … |
| maximal normal subgroup of | #72239476, #72278399, #72278497, … |
| maximal quotient of | #72264524, #72264633 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536564);