Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213736 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2061 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,22)(18,21)(19,23)(20,24);(1,3,4,7)(2,5,6,8)(14,21)(22,23);(2,6)(5,8)(9,11)(10,14)(12,19,20,17)(13,18)(15,23,24,22)(16,21);(9,12)(10,15)(11,17)(13,20)(14,22)(16,24)(18,19)(21,23)Polycyclic presentation:
2^9|b^2=f,c^2=h,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^c=de,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240905, #72266050, #72266052, … |
| maximal normal subgroup of | #72240905, #72266050, #72266052, … |
| maximal quotient of | #72240915, #72302092 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289898);