Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #612424 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(10,13)(12,15)(17,18,23,21)(19,24,22,20);(6,8)(9,10,14,13)(11,15,16,12)(17,19,23,22)(18,20,21,24);(5,6)(7,8)(9,11,14,16)(10,15,13,12)(17,20)(18,22)(19,21)(23,24);(1,2)(3,4)(9,12)(10,11)(13,16)(14,15)(17,21,23,18)(19,24,22,20);(9,13,14,10)(11,12,16,15)(17,22,23,19)(18,24,21,20)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fg,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch116 conjugacy classes — view the full character table.
| maximal subgroup of | #72305704, #72305715, #72305739, … |
| maximal normal subgroup of | #72305704, #72305715, #72305739, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 519740);