Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4622 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)(3,4)(5,6)(7,8)(9,10,13,12)(11,15,16,14)(17,18,20,22)(19,23,24,21);(5,7,8,6)(9,11,13,16)(10,14,12,15)(17,19,20,24)(18,21,22,23);(6,7)(18,22)(21,23);(2,4)(6,7)(11,16)(14,15)(19,24)(21,23);(9,12,13,10)(11,15,16,14)Polycyclic presentation:
2^9|a^2=fh,b^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=df116 conjugacy classes — view the full character table.
| maximal subgroup of | #72239620, #72265611, #72303391, … |
| maximal normal subgroup of | #72239620, #72265611, #72303391, … |
| maximal quotient of | #72239733, #72329843, #72336037 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 516409);